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Integrating stand and forest models for decision analysis Williams, Douglas Harold 1976

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INTEGRATING  STAND  AND  FOREST MODELS FOR DECISION  ANALYSIS  by  DOUGLAS  HAROLD WILLIAMS  B.Sc, Simon F r a s e r U n i v e r s i t y , 1970 M . S c , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1972  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS  FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  in the FACULTY OF GRADUATE STUDIES Department o f F o r e s t r y  We a c c e p t  this thesis the r e q u i r e d  as conforming t o standard  THE UNIVERSITY OF BRITISH COLUMBIA December, 1976 CcT)  Douglas Harold Williams  In p r e s e n t i n g  this  thesis i n p a r t i a l  f u l f i l m e n t o f the requirements  an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree the  library  s h a l l make i t f r e e l y  that  and  study.  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f the  thesis  f o r s c h o l a r l y purposes may by h i s r e p r e s e n t a t i v e s .  be g r a n t e d by  thesis for financial  written  permission.  The  University of B r i t i s h  Vancouver, Sanada V6T  April  1W5  29,  1977  Department o r publication  g a i n s h a l l not be a l l o w e d w i t h o u t  Forestry  2075 Westbrook P l a c e  the Head o f my  I t . i s understood t h a t c o p y i n g o r  of t h i s  Department o f  a v a i l a b l e for reference  Columbia  for  my  ii  ABSTRACT SUPEHVISOH: A. Models  of  forest  wide a c c e p t a n c e This  thesis  decision  each by  their  level, one  lands  planning  a planning  cr  the  more  use  process  problem  seme o f which a r e  planning should  chain.  be  capable  process.  optimized  a t each  Multistage  decision  was  The  At any  management.  chain  to  a n a l y s i s was  by  level,  problem  a c t i o n s on  a stand  used  a set of  other  not  to  combined  were  the  be  management levels  l e v e l s -of 'separated  examine  or treatment  the and  1  unit,  multistage allocation  dynamic  programming  and  stand  variables,  components  of  the  as  scheduling  management  and  allocating  constrained  unit. two  management a c t i o n s  on  structure  commodity  state  the  accomplished  underlying  management u n i t  approaches to o p t i m i z a t i o n of Scheduling  the  d e c i s i o n and  identified  c o m m o d i t i e s a c r o s s a management  best  characterized  a model of t h e d e c i s i o n  to the  should  and  s t r u c t u r e . At  neighbouring  c o n s t r a i n t f u n c t i o n s . The  combined  is  hierarchical  Columbia  level.  o b j e c t i v e and  were c o n s i d e r e d .  British  identified  of l i n k i n g  problem, i d e n t i f y i n g  Several  of t h e  constrained  mathematical s t r u c t u r e of the  serial  lands  gained  i n a framework amenable  management o b j e c t i v e s and  planning  unit  forest  Service involves a multilevel  problem  and  management u n i t s have  f o r planning  integrates  forest  decisions, of  tools  and  analysis.  The Forest  as  stands  KOZAK  through of  the  component c a n n o t recursions,  algorithms  due  a  treatment the  However,  the  v i a the  usual  optimized to  a  problem  that exploit  problem. be  level  continuous  and  i i i  multidimensional improves but, of  as  with  dynamic  states.  allocation  efficiency  functions  programming  discrete  with  regard  to handle  Conversely,  problem  subproblem  The  maximum to  approach  the s t a t e space  programming, i t r e g u i r e s d e c i s i o n i n v e r s i o n  transition  commodity  with  boundary  approximation  a  linear  of  model  i s very e f f i c i e n t ,  but  values the  and  of  the  commodity  o p t i m i z a t i o n by  t r e a t s the  stand  level  inadequately.  fitter the  space.  computational  the  linear  state  consideration  discrete  multistage  optimum  o f the  role  of  Lagrange m u l t i p l i e r s  f o r m u l a t i o n of the  approaches  were  problem, the  synthesized  through  in  linear  and  Dantzig-Wolfe  decomposition. The  general  problem  management a c t i o n s w i t h and  two  pattern  system  was  used  surface  of  search  technigue,  the  stand  that  optimized  were  tested  stand  models. The  with by  a  with  network linear  submodels, objectives.  model was through single  The  examined  conversational explore  optimum  was  sought  the by  a  algorithm.  embedded i n a n e t w o r k  formulation  dynamic  programming.  tree/distance  stand  decomposition.  The  a management u n i t conventional  of  simplex  model  under  a  was  to  sequential  management u n i t  with  model  interactively  model. the  o p t i m a l sequence  First,  f o r m u l a t i o n of the  Dantzig-Wclfe  demonstrated  a  an  simulation  considered.  Second, vas  a stand  finding  technigues  supervisor objective  of  Both  independent  model  (Timber  RAM)  present  85,000 h a . net  and  whole  was  combined  and  optimized  decomposition of  techniques  system  simulated  worth and  mixed  was by  8  goal  iv  The  decomposition  components: planning  an  system  existing,  i s optimized  system, with  and  estimates  decomposition  approach  forest  stand  linear  available  powerful  was  economically  programming  than  formulations.  accurate  actions.  attractive,  could  unit  mathematical  models p r o v i d i n g  o f r e s p o n s e s t o management  problems of f a r g r e a t e r complexity conventional  three  o p e r a t i o n a l model o f management  by a c o m m e r c i a l l y  programming detailed  combines  The  solving  be a t t e m p t e d  under  V  TAELE OF CONTENTS  ABSTRACT  i i  TABLE OF CONTENTS  v  L I S T OF FIGURES LIST  ,.viii  CF TABLES  X  ACKNOWLEDGEMENTS  xi  1. I n t r o d u c t i o n 1.1 M o d e l s  1  I n The P l a n n i n g  1.2 The P l a n n i n g Service 2. A Review  Process  Process  Of The B r i t i s h  Stand  2.1 S t a n d  Simulation  2.2 S t a n d  Optimization  And Management  Models  Onit  Models  ....  Models  Simulation  9 9 15  Unit  2.4 Management  Unit Optimization  Models  ..................  20  ............................................  23  3.1 T e r m i n o l o g y  Models  17  ................  3. P r o b l e m A n a l y s i s - The P r o c e s s  Of A b s t r a c t i o n  ..........  And N o t a t i o n  24 24  Land U n i t s 25 Management A c t i v i t i e s 26 S t a t e And S t a t e T r a n s i t i o n 27 Management O b j e c t i v e s And R e t u r n s . . . . . . . . . . . . . . 29 Management C o n s t r a i n t s . . . . . . . . . . . . . . . . . . . . . . . . 30  3.2 G e n e r a l  Problem Formulation  3.3 M u l t i s t a g e 3.3.1  Forest  ............................  2.3 Management  3.1.1 3.1.2 3.1.3 3.1.4 3.1.5  Columbia  2  3  Of F o r e s t  2.5 Summary  .....................  ........................  Analysis  MP 1 As A 2 - p o i n t  32 35  Boundary  Value Problem  .......  36  vi  3.3.2 MP2 As An I n i t i a l V a l u e P r o b l e m 3.3.3 MP2 As A S u b p r o b l e m Of MP1 3.3.4 The S e r i a l M u l t i s t a g e Model 3.4  ,  A p p r o a c h e s To O p t i n i z a t i c n  49  3.4.1 D e c o m p o s i t i o n Ey Dynamic Programming 3.4.2 The D i s c r e t e Optimum E r i n c i p l e 3.4.3 A p p r o x i m a t i o n H i t h A L i n e a r Model 3.4.4 D a n t z i g - W o l f e D e c o m p o s i t i o n .. 4. O p t i m i z a t i o n  4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6 MP2  ..........  49 54 59 63  .........................  67  ................................  67  Of The S u b p r o b l e m  4.1 D i r e c t O p t i m i z a t i o n  4.2  41 45 48  The S e q u e n t i a l S i m p l e x A l g o r i t h m . . . . . . . . . . . . . . 70 The O p t i m i z a t i o n S u p e r v i s o r Program .. 74 T e s t C a s e : Meyer's Model 79 T e s t C a s e : G o u l d i n g ' s Model ., 93 T e s t C a s e : K i l k k i ' s Model 104 Summary And D i s c u s s i o n 111 As A Network P r o b l e m  114  4.2.1  G e n e r a t i n g The G r a p h Of A l t e r n a t i v e Management Seguences ........................ 4.2.2 F i n d i n g The Optimum Management Seguence 4.2.3 L i m i t a t i o n s Of The D i s c r e t e F o r m u l a t i o n ....... 4.2.4 Summary And D i s c u s s i o n 5.  Joint  Optimization  5.1 The L i n e a r  Of 8E1 And MP2  Program  5.2.1  5.4  140  Model  143  D a n t z i g - W o l f e D e c o m p o s i t i o n W i t h MPSX  5.3 An Example Approach 5.3.1 5.3.2 5.3.3 5.3.4  V i a D e c o m p o s i t i o n .... 140  Master Problem  5.2 D e c o m p o s i t i o n Of The L i n e a r  122 129 135 139  .145  P r o b l e m To D e m o n s t r a t e The D e c o m p o s i t i o n i,  147  Problem D e s c r i p t i o n The U n c o n s t r a i n e d Optimal S o l u t i o n S t a r t - u p P r o c e d u r e And F i r s t F e a s i b l e S o l u t i o n Final Solution  148 150 156 157  t  A G o a l P a r a m e t r i c A n a l y s i s With The D e c o m p o s i t i o n Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  5.5 Summary 6. C o n c l u s i o n s  And D i s c u s s i o n  164  167 ., 171  vii  BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX 1  -  A Summary Of Symbols  APPENDIX  -  Iterative  II  Conditions APPENDIX I I I -  .........  183  ................................  185  And N o t a t i o n  A p p r o x i m a t i o n Of  Distinguishing  Final  P o i n t s On A S t o c h a s t i c 189  Surface APPENDIX  IV  -  Listing  APPENDIX  V  -  Simulated  APPENDIX  VI  -  .191  Of SIMOPT S c a n n e r Managed S t a n d Y i e l d  Tables -  Meyers APPENDIX V I I -  APPENDIX  VIII -  APPENDIX IX  -  APPENDIX  -  X  176  Simulated  ... Stand T a b l e s  - Goulding  .........  198 203  A M u l t i p l e D e c i s i o n T h i n n i n g P r o b l e m As A Network P r o b l e m ( K i l k k i ' s S c o t s P i n e Model)  206  A M u l t i p l e D e c i s i o n T h i n n i n g P r o b l e m As A Network P r o b l e m ( G o u l d i n g ' s D o u g l a s F i r Model).  218  A wolfe-Dantzig  MPSX  D e c o m p o s i t i o n Program I n  T r e a t m e n t u n i t Management O u t l i n e s F o r D e c o m p o s i t i o n D e m o n s t r a t i o n P r o b l e m .......  222 226  viii  L I S T OF FIGURES  1. L e v e l s o f f o r e s t 2. A c o n c e p t u a l  model  3. MP 1 as a 2 - p o i n t 4.  planning  MP2 a s an i n i t i a l  .  ........................  o f t h e BCFS p l a n n i n g  boundary value  value  process  ........  problem  6 40  problem  44  .5. MP2 a s a s u b p r o b l e m o f M.P1 6. E l e m e n t s o f a s i m p l i c i a l  4  47  search: r e f l e c t i o n ,  expansion,  and c o n t r a c t i o n  72  7. The s i m u l a t i o n c p t m i z a t i o n  s u p e r v i s o r system  ...........  75  8. S e n s i t i v i t y o f meyers' model t o r a n g i n g of t h e i n t e n s i t y of t h e f i r s t t h i n n i n g . 9. The pathway o f t h e o p t i m i z a t i o n a l g o r i t h m s on a s u r f a c e of p r e s e n t n e t worth ( K i l k k i ' s m o d e l ) .  89 107  10.  Volume/age c u r v e s c o r r e s p o n d i n g t o s i x l e v e l s o f d e n s i t y a t age 20 y e a r s , o f S e c t s p i n e ( K i l k k i ' s m o d e l ) . . . . . . . . . . . ... ..... 115  11.  A l t e r n a t i v e management c y c l i c d i r e c t e d graph.  12.  sequences represented  as a 116  A l t e r n a t i v e management s e q u e n c e s r e p r e s e n t e d as an a c y c l i c d i r e c t e d g r a p h o f s t a t e s and t i m e s t a g e s . ......  13. L a b e l l i n g program.  o f a r c s p r o d u c e d by t h e g r a p h  117  generating 124  14.  S t a t e ' d e f i n i t i o n and p r e c e d e n c e g r a p h o f a m u l t i p l e t h i n n i n g d e c i s i o n p r o b l e m ( K i l k k i * s S c o t s p i n e model) .. 125  15.  A management o u t l i n e r e p o r t f o r m u l t i p l e t h i n n i n g s o f / S c o t s p i n e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127  16.  A g r a p h r e p r e s e n t i n g t h e a l t e r n a t i v e management s e q u e n c e s o f a m u l t i p l e t h i n n i n g problem i n S c o t s  pine.  129 .  17.  An o p t i m a l management s e g u e n c e c a l c u l a t e d by embedding K i l k k i ' s S c o t s p i n e model i n a network f o r m u l a t i o n ....... 131  18.  S e n s i t i v i t y o f t h e o p t i m a l management-sequence t o d i f f e r e n t d i s c o u n t r a t e s ( K i l k k i ' s model i n a net-work formulation)  134  19. G o u l d i n g ' s model i n a network d e c i s i o n p r o b l e m , w i t h a s i m p l i f i e d d e f i n i t i o n o f t h e s t a t e . ....................  137  <  ix  20.  F l o w c h a r t o f t h e MPSX D a n t z i g - W o l f e decomposition program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  146  21.  Unconstrained  optimal  policies  f o r treatment  units  1-4  151  22.  Unconstrained  optimal  policies  f o r treatment  units  5-8  152  23.  Volume f l o w  plans  .... 155  graphs f o r three  management  unit  24.  Stopping r u l e - a s y m t c t i c behavior of the decomposition model o b j e c t i v e f u n c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158  25.  C o n s t r a i n e d o p t i m a l management u n i t s 1,2,4 and 5  sequences  Constrained optimal u n i t s 6, 7 and 8  sequences  26. 27. 28.  management  treatment 161 treatment 162  D e c a d a l volume h a r v e s t s computed by g o a l - p a r a m e t r i c programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ftrea  of unit  created  s u b j e c t t o t h e management  at each  sequences  decomposition  29.  Searching  f o r complementary  30.  Precedence graph  167  169 slackness.  for kilkki's  demonstration  ....-v.*. problem.  188  .. 216  X  L I S T OF  1,  2.  C o s t and analysis  TABLES  b e n e f i t assumptions used o f M e y e r s ' model.  i n the  optimization 81  C o s t and b e n e f i t a s s u m p t i o n s u s e d i n t h e e v a l u a t i o n o f G o u l d i n g ' s model ( a f t e r Hoyer (1975), page 10) . . . . . . . . .  •3. S e n s i t i v i t y o f t h e o p t i m a l two t h i n n i n g management seguence t o v a r i a t i o n s i n the i n t e n s i t y of the f i r s t t h i n n i n g . ( K i l k k i ' s model) . . . . . . . . . . . . 4.  5.  95  109  S e n s i t i v i t y o f t h e o p t i m a l 2 t h i n n i n g management s e g u e n c e t o v a r i a t i o n i n t h e age o f t h e f i r s t thinning. ( K i l k k i ' s model)  110  Optimum management s e g u e n c e s w i t h two c o m m e r c i a l t h i n n i n g s , at i n c r e a s i n g d i s c o u n t r a t e s . ( K i l k k i ' s model)  111  6. Computer c o s t s f o r t h e d e m o n s t r a t i o n p r o b l e m Dantzig-Hclfe decomposition  of 169  xi  ACKNOWLEDGEMENTS The for  author  wishes  h i s guidance  graduate  t o thank  h i s s u p e r v i s o r . Dr. A. Kozak,  and p a t i e n c e t h r o u g h o u t  studies.  Also,  the author  the author's  thanks  p e r i o d of  Mr. G.G. Young f o r  introducing  him t o t h e t e c h n i g u e s o f o p e r a t i o n s r e s e a r c h and  encouraging  him i n t h e w r i t i n g  Dr. reviewed Dr.  J.H.G. S m i t h , the t h e s i s  J.L. Clutter,  gratefully  friends  Forest  D r . A. Chambers, a n d Dr. L.G. M i t t e n  and p r o v i d e d welcome a d v i c e and c r i t i c i s m .  who.read t h e t h e s i s  the author  would  and a g u a i n t a n c e s  Service,  perspective  dissertation.  as e x t e r n a l examiner, i s  acknowledged.  Finally, his  of t h i s  like  h i sgratitude to  a t OBC and i n t h e B r i t i s h  who have a t t e m p t e d  of forestry  t o express  in British  Columbia  t o g i v e him a r e a l i s t i c Columbia.  1 1.  Introduction  Decision selects  analysis  specific  courses  courses of a c t i o n , In  the l a s t  research  to  analysis'.  five  from  of  years lands  this and on  shift  reflects  i s t h e r e d u c e d emphasis  a p p r o a c h e s . The n e x t  This existing  using  thesis tools  exploring  five  at  the  the  problems  operations  to  a  reality  that  only  new  This  definition  i s provided  by S.M.  the the  techniques  see i n c r e a s i n g i n planning  emphasis  systems.  for  and s i n g l e a  of  for  Another a s p e c t o f  on d e v e l o p i n g  techniques  subtle  'tools  integrating stand  decision  framework.  1  possible  recognition  decision.  in  of  undergone  making'  t h e management u n i t  planning  s e t of  one  1  years w i l l  some  which  his goals .  the e x i s t i n g t o o l s  presents  a  have  for decision  and  by  applications  manager c a n make a c r e d i b l e  effectively  and  the  'tools  This  shift  from  planning  l i m i t a t i o n s of the technology resource  action  i n order to achieve  forest  redefinition,  i s the a n a l y t i c a l process  Lee (1972, p. 3 ) .  level,  analysis  2  1,1  Models I n The P l a n n i n g The  use o f computer  management,  and  will  For  be d e f i n e d  uncertainties  this discussion,  t h e computer  provides involved  is  the  is  quickly  decision rather  the  an the  problem  i n mathematical in  a  computer  making  and  problem: ?  alternative  decisions  information  available  decisions  to  to  be  d e t a i l s . The  process  e v a l u a t e the  ?  problem of  The  modeller  while l e a v i n g out  abstraction  o f human problem s o l v i n g , and p e r c e p t u a l  problem:  by t h e b a s i c we d e a l  with  data  o f any b u t  stands, type  the  i s an capacity simplest  groups or f o r e s t s ,  trees.  a r e s u l t o f t h i s process of a b s t r a c t i o n , approximation elements  approximation. problem  different  f o r the complexities  or o b j e c t i v e s of  o f these  exhausted  than  As  of  i n a decision  range  many  i s t h e e s s e n c e o f model b u i l d i n g .  less relevant  only  o f t h e problem  capture the crux of the d e c i s i o n  part  for  that  ?  abstraction  integral  widespread  a model o f a d e c i s i o n  framework  what i s  implications  the  a  -  what  term  so  business  model.  what i s t h e o b j e c t i v e  considered  must  collective  -  -  is  research,  o f t h i s mathematical a b s t r a c t i o n  language creates model  a  planning  a s an a b s t r a c t i o n  terms. Encoding  1  models i n s c i e n t i f i c  government  • m o d e l i n g ' h a s become techniques.  Process  of  to r e a l i t y , the  model  Consequently,  rarely constitutes  a  the  and any e r r o r s will  result  model  i s  i n the choice  in  a  biased  s o l u t i o n t o t h e mGdel  decision  a s o l u t i o n t o the r e a l problem.  It i s  3  the  r o l e o f t h e manager t o t a k e  of  the  decision  management decision  model  model  is  Forest  of  the  Service  characteristics  The P l a n n i n g  (BCFS)  Process  Planning  planning  (Figure  will  provincial  included.  decisions,  to set  some  1)  ;  the  level  of  cut  not  of  reveal  other  Columbia  block,  the  lands  at each  objectives  which  the  Eritish  desireable  Forest  Service  four  l e v e l s of  watershed,  planning  level  and  a  are constrained  the  role  of  should  be  i s characterized set  of  by  management  by t h e  neighbouring  levels. example, t h e o b j e c t i v e s o f management f o r e s t resource  together are  and  with  major  stratification goals,  of  For completeness, t h e  forest  problem  at the r e g i o n a l l e v e l ,  system,  process  Of The B r i t i s h  management  ensure t h a t  zoning  analysis,  D i v i s i o n o f t h e BCFS r e c o g n i z e s  The p l a n n i n g  o r more  For  for  solve the r e a l  model.  management u n i t , and t h e r e g i o n .  planning  tool  planning  of a planning  The  one  a  analysis  making.  Columbia  the  i n the  and t o use them t o h e l p  p r o b l e m . The  Consideration  1.2  i n s i g h t s gained  the  production  layout  and use z o n i n g  example, t h e p r o d u c t i v e  the  goals  planning are  for  the  a r e met. Land s t r a t i f i c a t i o n  decisions  constrain  unit  and made  both  scheduling at  reflect  development  capacity  this  of  of each l a n d  of  level.  regional these unit  unit,  and use  the  road  The l a n d land  use  goals.  For  as  measured  F i g u r e 1.  Levels of f o r e s t planning,  (from BCFS (1975) ) .  5  by  the  annual  information The as  by  forregional  planning  a chain  level,  allowable  decisions  made  or  process.  Similarly,  the  each  While  capacity  level,  fiscal  and  the  with  the  varying  province  decisions  to  planning  of the f o r e s t lands  decisions  f o r optimum  multilevel planning  problem  allow  must be detail.  a n d t h e market  a t each  at  each  level  level  of  allow  planners  at  F o r an e x p l a n a t i o n H a l e y (1975).  each  level  the  t h e bounds o f  level.  structure  of  the  (operational)  (e.g.  'degrees  use o f s p e c i a l i z e d o r  planning  d e c i s i o n s t o be t i m e s t a g e d .  conditions  are highly  These  d e c i s i o n making, w i t h i n  at that  be made a t t h e l o w e r  changing  of  addition  c e r t a i n 'state '  transcend  although  state  planning  d e c i s i o n problem  allows  constraining  In  seme d e c i s i o n s c a n be made f r e e l y .  knowledge,  The  planning  or constrained  These  levels,  requirements  freedom' i n t h e d e c i s i o n  opportunity  2  each  process. many  constrained,  local  At  figure.  planning  f o r f o r e s t products a f f e c t  planning  i s necessary  by a r r o w s between t h e r e c t a n g u l a r  the  productive  at  levels.  accompanying  between  considered  potential  the  information  The  2  above c a n be c h a r a c t e r i z e d  (Figure 2 ) .  other  are represented  links  factors  can  problems  at  spaces i n  decision  the  described  seme o f t h e d e c i s i o n s a r e l a r g e l y d e f i n e d  decision  of  (AAC) c a l c u l a t i o n  decisionmaking.  process  of planning  relationships  the  cut  short  term  problem  Planning  levels  in  o f t h e BCFS a l l o w a b l e  cut  decisions  response  market t r e n d s )  also  to  as long as  calculation  see  long range m a r k e t ft potentials,  philosophy of u s e , etc.  Province  boundary  each  rectangle  decisions  for  conditions  represents a  planning  the  set  of  management  l e v e l  Region most  decisions  l e v e l s  Management  of  are  constrained  by  other  management  t7  Unit  only  some  decisions  are  f r e e  at  each  l e v e l  Development Area or Treatment Unit  Cut  Block  4.  biophysical current etc. Figure  2.  A  conceptual  model  of  the  planning  process.  realities,  markets,  -  boundary conditions  7  they  stay  the  adjacent  decision  within  t h e d e c i s i o n and s t a t e c o n s t r a i n t s  levels  that  of  decision  leads  decisions  t o another  problem  One making  changing  conditions  societal  structure  values,  b a s e . The n e s t e d adjusting changing  or  of  management  up  and  down  the  i s no  planning  responsive  a to  such as markets,  i n the f o r e s t  problems  p r o c e s s i s t o show t h a t (flexibility,  structure rather  decisions .  As  3  abstraction*,  we  see  i s  chain  in  land  constantly response to  of the planning  planning  alternatives. a t any l e v e l  to  the  other  step  a planning  in  implies  the  model s h o u l d  that  nature)  of f o r e s t  land  'process  of  r e f l e c t the  as w e l l a s c o n s i d e r i n g  o f the system levels  dynamic  than the philosophy  system This  many o f t h e q u a l i t i e s o f  responsiveness,  a .first  that  structure  link  and  changes  planning  There  must be t o c r e a t e  flexible  catastrophic  system  process  from  problem  is  chain.  p u r p o s e o f t h e above h i g h l y s i m p l i f i e d c h a r a c t e r i z a t i o n  ideal  derive  of planning  an  multilevel  overall  o f the d e c i s i o n environment  decisions  the planning  the  that  in  conditions.  The of  results  process.  s o l u t i o n to the  o f the o b j e c t i v e s  radical  up and down t h e p l a n n i n g  model o f t h e BCFS p l a n n i n g  decision  of a  i m p l i c a t i o n of the nested  instantaneous or simultaneous problem.  The n e c e s s i t y  v i o l a t e s one o f t h e s e c o n s t r a i n t s  adjustment of planning This  planning.  i m p o s e d , by  all  the  a model o f a d e c i s i o n  must have t h e  of the planning  capability  process.  The  to  irodel  3 The a u t h o r a p p r e c i a t e s t h a t p l a n n i n g i n t h e BCFS may n o t o c c u r i n p r a c t i c e a s o u t l i n e d above. The d e s c r i p t i o n i s i n c l u d e d as an i d e a l i z e d example.  8  should  not  be  'separated'  and  optimized  in  isolation  at  each  level. To  optimize  of  the  the  be  considered  optimal  planning  over  as  throughout  level the  constraints.  effects  d e c i s i o n model i n t h e  d e c i s i o n and To  choose levels,  of  decisions  the  of p l a n n i n g  state linkages  decisions  planning  context  the  that  solution to  p r o b l e m s , and  be  can are  technique  transmitted  f o r each l e v e l  to  planning l e v e l  as  i t s d e c i s i o n s i n response. forest  isolated  solve  land  subsystem  multiple  consideration present  of  planning of  the  level  models  planning  models  pertinent planning  recognize explored,  the  analytical  multilevel  Specifically,  will  analysis.  and  be  treat  nature  models o f  a  process,  problems  a f u r t h e r a n a l y s i s of the  Planning  stands  process,  the c h a i n  Most an  single  multiple  must a l l o w t h e  adjust  the  forest  land  techniques  forest  attempt  simultaneously.  models,  of the  or  this  that  planning  will  problem.  explicitly  problem  management  after  thesis  planning  to  will  units  be and  i n t e g r a t e d i n a framework amenable t o d e c i s i o n  9  2• 1  Review Of F o r e s t  In be  Chapter  reviewed  the  provide  about  amount  detailed  will  Unit  and management  the v a r i o u s  and  a planning  models  simulation  And Management  2, f o r e s t s t a n d  to identify  examine  The  Stand  nature  lines of  Models  unit  models  will  o f d e v e l o p m e n t , and t o  the information  that  they  categories  of  problem. be  grouped  and o p t i m i z a t i o n .  into  Simulation  descriptive information  the  models g e n e r a l l y  about a s i n g l e p o i n t  provide  i n policy  space, i . e . the i m p l i c a t i o n s of a s i n g l e p o l i c y are predicted i n detail. about of  Optimum  seeking  t h e whole  some  and  of  may  2.1  Simulation  Stand  independent, require  The  models,  relative  distinction  simulations.  line  models a r e commonly distance  whole s t a n d  optimization  >  subdivided  work  of  models. D i s t a n c e  Newnham of  Douglas-fir  into  dependent, s i n g l e t r e e  independent  o f development  tfewnham's  between  Models  f o r distance  seminal  information  p o l i c y i n terms  models i s b l u r r e d : an  repeated  single tree and  The  the l o c a t i o n of the i n d i v i d u a l t r e e ,  unnecessary  active  involve  simulation  categories:  performance.  optimization  procedure  Stand  provide  p o l i c y s p a c e - what i s t h e b e s t  criterion  simulation  models  single  three  distance  dependent  while  models  location  is  models. (1964) tree  (Pseudctsuga  has r e s u l t e d distance  i n an  dependent  menziesii  (Mirb.)  10  France)  model s i m u l a t e d t h e d i a m e t e r  spatial  characteristics.  reduced  t h e g r o w t h . The  unmanaged  model was  the  output.  lee  revenues,  results.  and  Bella  (Michx.))  of  as dbh.  and  implemented  hemlock  Lin  ^Tsuga  it  (1969), Voss)  tree,  in  dependent model of  distance  Many  (197 4)  incorporated high  level  model o f  such  as s i t e  the into  ideas the  of u t i l i t y  by  dbh  of  demonstrated  system  to  and system Arney  a  by  slauca  for  of  thinning  Arney  (1974) and  Douglas-fir simulates r a d i a l  as  a  modelled  contact.  and  (1972) Hegyi  single and  management  silvicultural pruning.  described  whorl  Mitchell  species  s i m u l a t e numerous  concepts  single  (Moench)  was  mixed  as  Western  generalized  designed  alteration,  the  index  the  growth. C o m p e t i t i o n  was  cu  tremuloides  growth o f  structure  have c r e a t e d  system  volumes  limits  of a d j a c e n t t r e e s u n t i l  dependent  facilities  of  dbh  branches  has  was  pine  competition  White s p r u c e j P i c e a  h e i g h t and  Monserud  treatments,  model  t o compute  spatial  m o d i f i c a t i o n of the  by g r o w i n g  and  a  Sarg.).  uneven-aged s t a n d s . The tool,  a  f o r Lodgepcle  p r o j e c t e d h e i g h t growth  h e t e r o p h y l l a (Kaf.)  involved  Ek and  s p a c i n g , and  others  information  (Populus  of the  Mitchell's  directly  aspen  trees  framework,  put c o n f i d e n c e  model  distance  known  approach.  the  model  facility  to  (1969) d e v e l o p e d  A substantial tree  added t h e  modelled  irregular  of the  improved  the  attempted  (197 0)  well  of  f o r normal d e n s i t y ,  conceptual  and  calibrated  JPinus contorta Douglas), and  the  model's components (1967)  calibrated  the g e n e r a l i t y  Newnham e s t a b l i s h e d  modified  a stand  Crown c o m p e t i t i o n of n e i g h b o u r i n g  stands, l i m i t i n g  After  growth o f  and  above  were  brought  to a  (1974).  Arney's  growth, e n a b l i n g  the  11  simulation adapted  of  t h i n n i n g and  Although  is,  of  the  information detailed  i t  produces.  of a  policy  dependent  on  indices  dependent  The  ccst within  line  independent  and  produced  stochastic  of  finding  and  guality, easy  the  models, i t quality  of  allows  estimates  of  conversion  of  facilitates  financial  of  of  which t o e v a l u a t e  single of t h i s  tree  the  meaningful!  t h e n e c e s s i t y o f stem  c h a r t s as  computer  of  processing  a  distance  approach are  biologically  development  Goulding  spacing  - an  problem  model a s  the  input, spatial  stands.  acceptable  and  and  disadvantages  Paille  basic unit.  thinning  Lamb,),  of bole form  i n c h , e t c , The  as  of clear  (1970)  independent of stand  the  nature  simulation  models i s not  models.  as  the  i n f o r m a t i o n with  in  of  stand  Douglas-fir  in  advantage  competition,  high  The  banksiana  Hegyi  management a c t i o n s .  products  detailed  inherent  relationships  shape.  stand.  models.  the  JPinus  bole  t r e e d i s t a n c e dependent  per to  on  d e s c r i b e Arney's  log grades  i s the major  of  to  The  rings  and  difficulties  the  risky  characteristics  Accurate  and  be  representative  gravity,  evaluation  pine  art of single  information  specific  stand  of c o m p e t i t i o n  fertilization  i t would  at l e a s t ,  stand  effects  A r n e y ' s model f o r J a c k  including  state  the  "single as  simulated  d e n s i t y , with  over  (Munrc,  a  distance  the  distance  with  (1972) s i m u l a t e d  results  tree,  wide  diameter the  range  Mortality  awkward, i n e l e g a n t , but  effective  ( b a s a l a r e a ) among d i s c r e t e  a  continuous  units  (stems).  single  of  tree  D o u g l a s - f i r growth  1973).  allocating  growth  measure Stage  of  was  sites,  partially  solution  to  of m o r t a l i t y  (1973)  used  a  12  similar  approach.  Clutter  and  Allison  (1974) d e s c r i b e d  a model t h a t  b a s e d on a s i n g l e t r e e jaer s e , b u t i s t h e l o g i c a l of  the approach.  Allison the  approximated  Weibull  then  into  the diameter  but a v o i d s  Eurkhart  The p r o b a b i l i t y d e n s i t y  i n terms of  preserves  and S t r u b  pine  diameter  d i s t r i b u t i o n with  expensive require  JPinus taeda  tree to  stem  developed is  a beta  distance  run  than  distance  The  fact  that  a s an o p e r a t i o n a l  concept  of  tables  regression Meyers  Clutter  stand,  used  empirical  regression  tree  list.  but approximated the  are  function. much  produce l e s s  and  less  models and dc n o t detailed  A l l i s o n ' s model was  Zealand  Forest  distance  Products  independent  prediction  tools,  volume-age c u r v e s ,  s y s t e m s t o be c o n s i d e r e d  by t h e U.S. F o r e s t pine  classes.  of the single  dbh  and  o f the approach.  (1971) h a s p r o d u c e d  manage p o n d e r c s a  acre  diameter  dependent  t o o l by New  whole  function-based  operationally to  or  per  models  but g e n e r a l l y  e n c o m p a s s e s a wide r a n g e o f y i e l d normal  area  probability density  independent  evidence of the u t i l i t y The  area  L.) p l a n t a t i o n s ,  maps as i n p u t ,  information.  p r o b a b i l i t y . Basal  (1974) u s e d a s i m i l a r a p p r o a c h t o model  Loblolly  Single  was  basal  a lengthy  with  function  much o f t h e d e t a i l  maintaining  C l u t t e r and  d i s t r i b u t i o n o f t h e stems  a r e a l l o c a t e d among t h e v a r i o u s  formulation  extrapolation  a dbh l i s t ,  25 c l a s s e s o f e g u a l  and m o r t a l i t y  stems p e r a c r e  approach  of maintaining  distribution.  divided  increment  This  Instead  i s not  (Pinus  functions  to  whole s t a n d Service  including  as well  as the  here.  models t h a t  a r e used  i n Wyoming and C o l o r a d o  jgonderosa L a w s , ) . predict  models  dbh,  Meyers'  estimate  model height.  13 estimate  height  predict for  managed s t a n d s  cutting  after  is  stand  basal  area  quality. goals.  average  'Best' stand  E n t e r i n g the curve  with  yield  with  facilities  seed  tree,  o f the stand  diameter  to  evaluate  to  Increment per  year  model,  functions  with  and u n t h i n n e d  table  projection.  site  production  to stand  growth  management The  by  financial periodic  earned.  d i s t a n c e independent management  practices.  and c u b i c f o o t form  model  volume  the b a s i s of  t o Goulding's  the  simple  and Moser  (1974),  average  example o f t h e u s e o f a r e l a t i v e l y management d e c i s i o n s .  (1974) d e s c r i b e d s y s t e m s o f e q u a t i o n s  i n diameter  to the diameter  Allison  desired  Hcyer's  s i m u l a t i o n model t o a n a l y s e  and  of  model.  stand  similar  systems.  stands  compare c l o s e l y  s i m u l a t i n g growth  and  thinning returns  timber  gross basal area  i s an e x c e l l e n t  for  after  intensive  study  (1974)  area  reports detailing  a whole s t a n d  of  f o r thinned and r e s u l t s  Ek  graph  to r e f l e c t  simulate  i s simulated,  Douglas-fir  each  per acre.  or c l e a r c u t t i n g  (1975) used  after  basal  for local  a diameter  c o s t s and r e t u r n s , and a n a l y s e s o f r a t e Hoyer  predict  o f t h i n n i n g s t u d i e s and d a t a  to construct a  stand  to  t o be l e f t  between  and  applicability  algorithm  density  Results  thinnings,  (1973) model uses t h e same a p p r o a c h  the  shelterwocd,  an  density i s selected  a p p r o p r i a t e b a s a l area  but  from  The model's  by  Stand  p l o t s a r e used  over  Meyers'  mortality.  as a r e l a t i o n s h i p  diameter.  temporary  increase  enhanced  cutting.  i s expressed  average from  diameter  non-catastrophic  stocking  the  and  classes.  distribution  This approach  f u n c t i o n model  but i s i n c l u d e d here  of  as a method  i s very Clutter  f o r stand  14  Of stand  the t h r e e  projection  little  more  than  c a t e g o r i e s of stand  models r e v i e w e d ,  approach  the  requires  conventional  least  expensive  i n terms o f  also  provides  the  individual their  t r e e data  to products  A  solution  processing  example,  Adams  to  more a p p r o p r i a t e compatible  to s e l e c t  period  of  level,  and c o n v e r s i o n  of  time  periods  (1976)  trading  off  approaches  o f a management  regime. For  suggested  using  approach  models a t a h i g h e r a  detail  to  level  proposed  use  single  calibrate  M e y e r s ' whole s t a n d  (1976)  simulations  management is  a  from  time t h e diameter c l a s s  of  detail for  diameter origin  structure manipulation  growth  the l e v e l  of  of  modelling  f o r p r o j e c t i o n s . Hegyi  i n the simulated  Another other  problem  Ek  that  system  However, i t  l a c k i n g . Management a c t i o n s and  at coarse  o f age) when  At  and i s t h e  f o r d e c i s i o n a n a l y s i s as  f u n c t i o n model t o p r o j e c t a s t a n d  interest.  user  the  and  (10-30 y e a r s  (usually  for appraisal i s difficult,  different  distribution  of  processing.  information  are t o t a l l y  data  statistics)  c o s t i s t o use t h e a p p r o p r i a t e  simulate  time  computer  responses are modelled  yields  to  least  inventory  least  t h e whole  appropriate  be  developing will at  a  allow the each  time  seguence.  use d e t a i l e d of  would  models would be  is  which  to a  models t o c a l i b r a t e  aggregation.  Stage  t r e e distance independent p r o j e c t i o n system.  (1973) model t o  15  2.2  Stand  Optimization  Some a t t e m p t s sequences through but  the  most  programming  problem stand  of  optimal  with  stand o p t i m i z a t i o n  the  state  that  timber,  simulates was  eguation  Schreuder  extremely f o r thinned  (1971)  dynamic  Naslund  (1969) e x a m i n e d t h e same principle, through  necessary  The  solution  policies,  H i s model  deterministic  simplistic,  consisting  growth  formulation  with  closely  an  to determine  f o r an e v e n - a g e d  analysis.  the  related Naslund  optimal  forest.  continuous to  dynamic  cnly  derived  policy  and d i d n o t  model  to  technique. programming  again  i s notable,  the f i n a n c i a l  transition,  increment  (1970) u s e d a dynamic  cutting  had  used a volume  problem  for  approach  stands.  schedules  multistage  DP  were  unthinned  technique  conditions  an e x p l i c i t  simulating  a  age. R e s u l t s  and  programming  t h i n n i n g and r o t a t i o n  function.  saw  a d v a n t a g e s . As i s t y p i c a l  and N e l s o n ,  optimal  optimum  of growing stock i n a  the  a  stage  the  but  increment  in  Kilkki  to  stand  Similarly,  present  levels  programming  is  o f saw  the  has i n v o l v e d dynamic  the  and  i.e.  programming  date  1964)  acre,  component  maximum  to  and N e l s o n ,  The model's s t a t e i s t h e volume o f  models, the  function  work  management  a rotation.  computational  o f a volume  a n a l y s i s (Chappelle  (1968) a p p l i e d dynamic  to Chappelle  growth  optimal  models.  determining  per  comparable  been made t o d e r i v e  marginal  and A k i n  during  timber  have  successful  (DP)  Amidon  Models  aspects  based  however,  for  of stand  optimization  on a volume  compute increment  i t s emphasis  on  management.  models  described  above  16  constitute have  a clear line  attempted  development  to  i n an  reported  existence  by  of  probabilities The  inch  dbh A  Eudra  t r a n s i t i o n f r o m one this  a tree  r e f i n e d by the  probabilities.  average  dbh,  variables.  predicting period,  as  the  trees A  and  average  age  functions and  of  method t o  the  be  ( 1 9 7 6 ) . These a u t h o r s a r e  matrix  is dbh  was  as  another.  and  by  In  facility  decision market  and  at the  density  yields  number o f  trees  fixed  to  two  of  the  were  state  models  with  at  treat  variables,  p r i c e as  end  Hool  Hcol's  to  the  time  equations a  10  year  beginning  modelled  planted,  is  methods,  c o r r e c t l y treated  density  6  formulation  (1975).  simulated  the  tree.  between t h e  g r o w t h was  the  c l a s s to  prepared  the  of  that  according  and  more  financial  volume  assumed  d i s c r e t e dynamic programming  diameter  The  the  oriented  state  density  Stand  stand  distributions  age)  the  Johnson  actions  Johnson  of  of  J o h n s o n added  diameter  a function  final  and  changes  finite  unbounded.  thinned,  Lembersky  stand  period.  regression  the  4 inch  processes  management  cn  Lembersky as  the  management  and  of  diameter c l a s s to  fundamental d i f f e r e n c e  Hool r e l i e d  horizon  of  The  of  of  assumption  from  Markov  Lembersky of  while  Markov  and  stand  intensities  that  (independent  moving  realistic  (1966) and approach,  nature  model i s b a s e d on  of  stochastic  with  (1S68), The  stationary  of  researchers  framework.  c l a s s , i s independent more  stochastic  a  of  A number of  model f o r p r e d i c t i n g d i a m e t e r  involving  the  the  optimization  implication  probability  development.  simulate  A Markov c h a i n was  of  with  number  of  harvested. reviewed the  first  was to  reported report  by  the  Adams and  Ek  optimization  17  of  a stand  model c f g r e a t e r  complexity  projection.  Deterministic  incorporated  into  formulation nonlinear  of  a  the  system  stand  was authors  converging  t o an  optimal  summary,  simulation models. detail  2.3  as  only  the  cf policy  the  extensive  as  that  growth, h a r v e s t i n g , and  described  used  C l u t t e r and  the  Forest  land  base and  cost Bamping  Operations  and  explored.  forest  stand  into optimization  sacrifice execution.  Models  forest the  stand  Harvard  simulation. Forest  and  p r i c e data  to  (1965) d e s c r i b e d  Simulator  (FOPS), but  and  unplanted)  used t o e v a l u a t e  Gould  and  simulator.  It  s t o c h a s t i c c a t a s t r o p h i c events  F o r m u l a s f o r p r e d i c t i o n o f growth  was  of  prepare  two  yield  were  system,  involving a larger economic  f o r the  provided  management  such  financial  a similar  more s o p h i s t i c a t e d b i o l o g i c a l and  (planted  be  nonlinear  not  modelled fire,  that a  in  management u n i t s i m u l a t i o n m o d e l s i s of  reviewed  projection  difficulty  should  forms  f o r speed o f  Simulation  on  resulting  and  (1965)  system  suggested  simplest  The  accuracy  simulation  literature  species  considerable  were  programming  gradient  ( G e o f f r i o n , 1970)  formulations  Management U n i t  reports.  a  s o l u t i o n and  0'Began  as  by  stand  models  problem.  have been s u c c e s s f u l l y i n c o r p o r a t e d  All  The  management  reported  formulation  class  mathematical  optimized  The  In  diameter  classical  algorithm.  decomposition  t h a n a s i n g l e whole  models.  appropriate as  policies:  input. an  The area  18  regulation  procedure  and  The  produced  extensive  program  policy  based  on  financial  economic  reports  maturity.  to  aid  in  analysis.  The  Purdue  educational simulation  Management  purposes  but  operational  unit  use  by  Planning  Game  insight  Meyers  into  (1970,  The  system  was  by  Edwards  efficiency  evaluated  savings i n cost  examples  systems are the (SIMAC)  Projection  of  (ACP)  of t h i s  for  kind  of  problems,  and  Timber  two-staged f o r e s t  time,  of  due  and  stands. reported  TEVAP r e s u l t e d to  in  increased  o p e r a t i o n a l management u n i t  simulation  management  of  plans.  Intensively  Managed  Sassman e t a l . (1972) and  system  Evaluation  i n v e n t o r y summaries  Application and  for  the  Simulating  system  designed  i n a 2-year o p e r a t i o n a l t e s t ,  (1973).  i n producing  Other  value  The  calculates  management o f even aged  substantial  was  management  1974).  prescribes  al.  the  1970)  s i m u l a t i o n models have been d e v e l o p e d  (TEVAP) s y s t e m  et  (Bare,  demonstrated  for developing  Management  and  regulation  o f t h e B.C.  Forest  Allowable  the  Allowable  Service  Cut Cut  (McPhalen,  1976) . The  systems  d e s c r i b e d above were d e s i g n e d  management. R e c e n t l y , to  aid  in  planning  Snohomish  Valley  Schreuder,  1976)  geographic  data  environmental decisions.  s i m u l a t i o n s y s t e m s have  The  to i n i t i a l i z e  the  integrated  Environmental uses  base t o  a  the  has  Network  been  of  forest  (SVEN;  timberlands constructed lands.  The  Bare  and  s e t o f s i m u l a t i o n models l i n k e d  examine  consequences system  use  for  of  the  alternative  substantial  resource data  physical,  base,  economic,  and  wildland  use  data needs: i n v e n t o r y and  to a  empirical  data  response  19  data  necessary to The  1976)  c a l i b r a t e the  Midlands  i s a system  assist  Resource  developed  a  timber  number c f m u l t i p l e  by  the  manage t h e i r  the  the  simplest  problem  information of  the  with  level  inexpensive  this  simulate  often  i n reports  crude  generally  methods,  are  type  and  of  Authority  w o o d l a n d s . The  schedule  which  will  to  system  produce  a  benefits.  Management u n i t s i m u l a t o r s but  (WRAP; Ha inner,  Tennessee V a l l e y  harvesting  use  models.  A l l o c a t i o n Procedure  p r i v a t e landowners to  constructs  subsystem  r a r e l y model s t a n d  However, t h e  financial  handled  detail  in  and  summaries can  of  stand  to  run.  the  detailed  output  any  components  of  the r e s u l t i n g Because  these systems  Conseguently, choosing  by  voluminous.  simulation,  model r e g u i r e s  examining  be  and  growth  decision sets of to  analysis  policies  decide  are  on  to the  •best' p o l i c y . If easily  the  definition  expressed  optimizing  in  procedure  of  what c o n s t i t u t e s a  functional is  attractive.  form,  then  'best' a  policy i s  searching  cr  20  2.4  Management U n i t O p t i m i z a t i o n  Optimization rapidly and  (1962) p r o p o s e d  early  study  of f o r e s t  attempts  usually than  computer  decomposition  facilities  by  overcome  these  Nautiyal programming optimum period is  unit  in  state.  model t o a n a l y s e  to  timber  to  Tcheng  (1966)  model,  Linear the  These  p r o b l e m s were rather  programming  management  unit  ( 1 9 6 7 ) , i n an a t t e m p t  •user  harvesting,  demonstrated  how  c a n be used t o s p e c i f y t h e from  an i r r e g u l a r  to s u s t a i n e d The  yield.  authors  to  economically  forest The  linear  during  'normal*  concentrated  the  forest  on u s i n g  the  e c o n o m i c r e l a t i o n s h i p s i n t h e management  problem. N a u t i y a l  compute  linear  were a p p l i e d  of harvests  target  of the  time.  and  programming.  management  processing  Pearse  techniques  planning  model  LP t o f o r e s t  described  difficulties.  of i t s c o n v e r s i o n  the  linear  and  pattern  public  (LP) model f o r  e t a l . (1966)  r e g u l a t i o n by l i n e a r  Liittschwager size  programming  Kidd  by t h e l a r g e s i z e  actual  problem  a linear  property.  at applying  limited  been  enterprises.  management o f f o r e s t case  have  and i m p l e m e n t e d o p e r a t i o n a l l y f o r many  private forest  the a  models f o r management u n i t p l a n n i n g  developed  Curtis  Models  (1966)  used  c o s t ' , a measure essentialy  the  same  of temporal  opportunity  linear  efficiency  cost  in  time  dimension. In linear of  the  another  p a p e r recommending  programming, post  evaluating  Navon and McConnen  optimal  forest  the a n a l y t i c c a p a b i l i t i e s  technique  (1967)  described  of parametric  management p o l i c i e s .  the  of use  programming f o r  This technique  allows  the  21  systematic of  the  exploration  optimum  One  of  the  first  industrial  (1971),  The  from  earlier  the  LP  matrix  cash  MAX-KILLICN  matrix,  Navon  selects  the  LP  the  the  RAM  generator  processed  by  the is  used  Timber  for  1 9 7 5 ) . An  extension  use  transportation W e i n t r a u b and  Navon  model has  of  linear  the  necessity  developed  s o l u t i o n system set  of  (Fortson,  'solves'  management  1970),  policy  and  performs  produced  Resource A l l o c a t i o n  Method  (RAM).  u s e s volume and  financial  IP  system.  the  U.S.  by  the  B.C.  of  the  timber  and  A  by  Forest  management been  Timber has  (Williams  recently  are  displays  and  model  to  timber  model  graphs.  Service  The  tables  writer  Service  public  for  linear  report  Forest  has  yield  management a c t i o n s  alternatives  to  RAM been  et a l . , include  described  by  attractive  for  (1976).  at  models have p r o v e n most the  recognized  model a s to  Clutter  for  by  programming  analysis  was  for  Baraping, 1965)  optimum  a  activities  linear the  system  p o l i c y i n t a b l e s , summaries and  proposed  decision  LP  and  system  a commercial  extensively  Linear  Ware  similar  management  optimal  by  optimum  generate a l t e r n a t i v e sequences of The  neighborhood  system. created  timberlands,  classes.  the  used e x t e n s i v e l y  ( C l u t t e r and  the  cn  be  1968)  p r o g r a m , PROPHET  (1971)  matrix  (Clutter,  A commercial  analysis  and  to  developed  FOPS s i m u l a t o r  A third  flow  models  f o r e s t s was  and  alternatives. a  IP  MAX-MILLION  g e n e r a t e s an  p o l i c y space i n  policy.  managing  the  of the  management u n i t inadequacies.  applied  constrain  to  The  management  volume  flow  level,  although  most s e r i o u s unit  planning  to o b t a i n  the  fault is  realistic  22  s o l u t i o n s , fin o p t i m i z a t i o n model l i m i t e d demand  curve  would  be  more  However, t h e demand c u r v e  by a  realistic  approach  downward  in  many  sloping  situations.  destroys the l i n e a r  structure  of the model . 4  ialker the  (1976) p r e s e n t e d  Economic  essentially maturity. sloped  Harvest  marginal  periodic  Optimization  to  linear  Model  A  sloped  cost  binary  demand  curves search  are  curves used  technique  to  that  and  Park  (1975)  financial  and/or  positively  calculate  i s used  described  timber  t o i d e n t i f y the  Walker's  holds. solution  technigue  a s ' r a t h e r ad h o c ' , and r e f o r m u l a t e d t h e p r o b l e m  optimal  control  principle. excellent control  demonstration  a  implicitly maintains Nazareth this  of the l i n e a r  (Johnson  linear considers  was b r o u g h t study,  and  o f t h e mathematics o f  model i s t h a t  alternative  programming  only a  management  1974).  model  that  seguences  but  i n t h e LP m a t r i x .  very s i m i l a r  sequences (1973)  management  to the author's a t t e n t i o n  small  Nazareth  decomposition  a l l feasible  is  maximum  models.  and Scheurman,  only a small s e l e c t i o n  discrete  as an  f o r o p e r a t i o n a l use b u t i s an  management  of a l l the possible  be c o n s i d e r e d  described  of  criticism  the  of the a p p l i c a t i o n  systems to f o r e s t  proportion  utilizing  The model i s t o o s i m p l e  Another  can  problem  is  of  h a r v e s t volume where t h e o p t i m i z a t i o n c r i t e r i a  HcDonough  models,  (ECHO)  an e x t e n s i o n o f t h e Faustman c r i t e r i a Negatively  harvests.  an a l t e r n a t i v e  The  work  of  during the course  t o elements  o f t h e model  * The demand c u r v e c a n be i n c o r p o r a t e d i n t o the l i n e a r model directly with separable programming, but with susbtantial computational burden.  23  described  2.5  i n Chapter  lands  levels  are  planning well  simulation  models o f  detailed  descriptive  policy. simplest for been  which were d e v e l o p e d  independently.  Summary  Forest unit  5,  Stand  models a t t h e  developed  various  with  model) t o c r e a t e  a powerful  tool  decision  p r o b l e m s . However, o n l y  are  to g e n e r a t e field  yield  i s ripe  operational.  a  Stand  provide  management only  the  descriptive information  systems  models  (usually in a  for analysis of the  can  stand  Management u n i t s i m u l a t i o n  optimization  management  g e n e r a l l y employ  simulation, s a c r i f i c i n g  combined  The  about  optimization technigues  stand  widely  and  degrees of aggregation  information  ease of computation.  used  and  stand  simplest  have linear  management  stand  information.  f o r s y n t h e s i s and i n t e g r a t i o n .  unit  simulations  24  3»  3.1  Problem  A n a l y s i s - The  Terminology  Various describe forest terms  will  Resource notation approach be  assigned planning  Notation  terms  are  proposed  throughout  planning  problem.  In g e n e r a l ,  f e l l o w t h a t o f t h e Timber by  Hilliams  Planning will  this to  and  rely  Beightler  as needed point  A  for  BAM  the l i t e r a t u r e  on  prepare  way  the  paper.  casting  a r e summarized  1971)  as  Assisted  derivative  notation  will  5  must be d e f i n e d  for  the  programming  constrained  i n m a t h e m a t i c a l programming  notation  Computer  mathematical  throughout the  the  BCFS  (Navon,  (1967). Terms and  some b a s i c t e r m s  models o f  to  t h e use c f management  system  the  p r o j e c t . The  generally  problem  s Symbols and  et a l  (CABP)  cf Hilde  introduced At  Abstraction  t h e e l e m e n t s o f m a t h e m a t i c a l programming  lands  modified  And  P r o c e s s Of  the  notation  forest  format.  i n Appendix  and  I.  lands  25  3.1,1 Land  The the The  forest  Yield  to  some  soil-landform island  Unit  management  area  treatment  Farm L i c e n c e that  and  formed  i s an a g g r e g a t i o n  should  treatment the timber  model  unit  area  to  be  a  Public  homogenous  such  as f o r e s t use.  by o v e r l a y o f type  management  be  o f management  are  with cover,  The  type  mapping.  islands  on t h e  characteristics.  be s u s c e p t a b l e  i s the area  island.  (TFL).  designated  and i s u s u a l l y  units  would  variables,  similar  forest  unit  are land units  levels:  and t h e t y p e  total  management  having  three  In the context  (PSYU) o r T r e e  unit  at  t o t h e same  management a c t i o n s . I n t e r m s o f t h e s t a n d  the  simulation  range model,  o f l a n d on w h i c h t h e management  r e s o u r c e c a n be s i m u l a t e d w i t h  one s e t t i n g  of  the  parameters. Let  treatment  i  be unit  a  type  island  or aggregation  i' C'-E  The  a  unit,  the  analysis.  classification,  treatment  Hence,  simply  descriptive  i s contiguous  A  be c o n s i d e r e d  the treatment  Columba,  tjjge i s l a n d s  respect  will  to the planning  British  The  of  unit,  m an a g e me n t una, t i s  Sustained  of  land area  management  subjected in  Units  partitioning  of E into  specific  l a n d management  methods  of  of type  treatment model,  to  form  described  by  Williams  islands  unit  E , and u a  i n E, Then  the  units that i s optimal  can  and Yamada  desirable  management  u C E  , i  Williams  in  treatment  be  (1976).  approximated  by  fora the  Computer a s s i s t a n c e i s  units,  A  typical  example  and Yamada i n v o l v e d t h e p a r t i t i o n i n g o f  26  2441  type  3.1.2  islands  into  Management  87 t r e a t m e n t  Activities  Management a c t i v i t i e s intervals  are  assumed  to  occur  in  discrete  of time.  Let  the  intervals  planning  (t  necessarily into  units.  =  horizon  1,2,...,  of  egual  a s e t of d i s c r e t e  T)  be p a r t i t i o n e d where  length.  This  intervals  the  into  T  intervals  partitioning  discrete are  not  of the horizon  c r e a t e s t h e time frame  of  the  system. The two  activity  on a t r e a t m e n t  l e v e l s , management a c t i o n s A  a  management  management  treatment  unit.  clearcutting action  action  are  example,  management  a t t i m e t cn t r e a t m e n t  activities  feasible  and management  i s any a c t i o n  For  unit  that  i s described at  sequences. c a n be a p p l i e d  planting,  actions.  Let  unit  Then  u.  spacing  over and  be a management the  set  of a l l  on u a t t i s A . . ut  A  management  extending management  s e q u e n c e - i s a s e g u e n c e o f management  o v e r t h e whole o f t h e p l a n n i n g  Let  s e q u e n c e and A^ be t h e s e t o f a l l f e a s i b l e  s e q u e n c e s cn t r e a t m e n t The  horizon.  set  Au  in  may  unit  sequence i s a  (k) u  u  be  a  management  u.  be r e f e r r e d for  a  actions  u.  The  t o as t h e s e t o f k  th  alternative  alternative management  27  3.1.3 S t a t e  The  And S t a t e  Transition  management  descriptive  state  variables  which  necessary  to  commodity  state  represents  commodity  which  must  For  time  problem  period A  stand  approximation  simulated the  how  actions,  management  then state  age  and  decision The  control  information  the s t a t e  variables  of  some  in  is  list  of  unit.  forest  a  lands  mathematical  A t any i n s t a n t  unit  as d e f i n e d  abstraction  For  simulation  diameters  t h e age  provide  sufficient  by  the  example,  in  model,  the  at breast  i s that  of  of the  approximates  unit.  stand  stand  The i n f e r e n c e  of a treatment  ccnstrained  change i n r e s p o n s e t o  state  treatment  independent  a  A  cut i n a particular  time i n t e r v a l .  simulation  the  decisions.  t h e management  commodity  model  and  height  diameter  information  making.  simulation and  model c o n s i s t s  Decision  v a r i a b l e s c a n be m a n i p u l a t e d adjust  f o r e s t land  decision  of a f u n c t i o n a l  state  the  the  constraints).  between  variables  across  simulation  this  the s t a t e .  distribution for  cf  volume o f t i m b e r  i n a given  Goulding's distance  comprise  status  ccnstrained  i s a set of  management  v a r i a b l e s . I f t h e model i s a v a l i d  system,  stand  and f u t u r e the  the  a l l  unit  t i m e , t h e model i s i n a p a r t i c u l a r s t a t e  state  real  of  treatment  be a l l o c a t e d  (i.e. yield  forest  management  is  a  provide  make i m m e d i a t e  example, a t y p i c a l l y  planning  of  in  and model  freely  r e s p o n s e t o changes  simulation  management a c t i o n s .  variables,  while  relationship parameters. the  state  in the decisions. For  model, t h e d e c i s i o n  The model p a r a m e t e r s c a n be  variables thought  are the of  as  28  constant  state  simulation As  and  (e.g. s i t e ,  a  predicts  state the  have f i x e d type,  future  states  where  v  _  L F C  is  u,t+l  the  unit  u i s M^,  } = M ({v u  vector  .c  ut  management  produced  similarly  of  (c  l u t  interest.  level  appropriate values  model  o f the c u r r e n t  state  arrays  > c  u f c  .  parameters  then t h e p r e d i c t e d  state  i s  ),a  ut  )  (3.1-1)  ut  state  and c . i s t h e amount o f  will  t , .. ., j c  2 u  The  subscript. c\  with  ut  u t  management  be m u l t i - d i m e n s i o n a l .  higher  simulation  o r consumed. The commodity  of J elements  commodity  the  a  commodities  into  a  throughout the  classification).  I f t h e model, c a l i b r a t e d  ut  commodity  values  as a f u n c t i o n  t+1 due t o management a c t i o n  u,t+l  soil  transition - function  to treatment  {v  the  that  species  management a c t i o n s .  appropriate at  variables  input-output  ) representing state  is  a  the J  variable  may  The c o l l e c t i o n o f t h e v a r i a b l e s  be s i g n i f i e d  by  the  omission  F o r e x a m p l e , c i s t h e JxOxT  of  matrix o f  29  3.1.4  Management O b j e c t i v e s  In  a m a t h e m a t i c a l programming  precisely unit  stated.  are  to  forest  Typical  maximize  minimize c o s t s . in  And B e t u r n s  problem t h e o b j e c t i v e  management volume,  objectives  maximize  f o r a management  present  The e c o n o m i c a s p e c t s o f v a r i o u s  management  were  reviewed  by  must be  n e t worth, o r  objectives  Bently  used  and T e e g a r d e n  (1965). A cost on both  a  treatment  was t a k e n .  be t h e p r e s e n t  R  i s associated  unit.  t h e management  action will  or p r o f i t  This  action  with each  action  and t h e  return  function  net value  function  where P i s t h e n e t p r o f i t  cf applying  unit  f v .,c^  u  while  in  state  ut  L  ut  the  action  sequence r e t u r n  returns  encountered  R  = R (v , u  The simply  objective  u  u  u  function  ' 0  the  this  work  (3.1-2)  )  a^  treatment  ] , 8 i s the discount  r a t e , and  J  B  over.  i s formed  i n a management  u  t  when  to  by  summing a l l  seguence.  I R  ,a ) =  c  Y ( t  action  y ( t , t ) i s number o f y e a r s t o be d i s c o u n t e d A management  state  used t h r o u g h o u t  ,a .) / ( 1 + e ) ut ut  action  i s a function of  t  management  The r e t u r n  = R ( v .c .a ) = P ( v ut ut ut ut ut ut  B  management  t =  ^  (3.1-3)  u t ;  f o r t h e whole management  t h e sum o f a l l t h e management  seguence  returns.  unit U i s  30  R  =  R(v,c,a)  =  u=l  3,1.5  Management  (3.1-4)  R  E  v  Constraints  Management  constraints  can  be  grouped  the  allocation  into  three  categories:  restrictions consumption) management  of  -  the  simple  first  categories For  constraints yield  of 1  usually  regulation  management  or  derived  the  states  intensity  the  seme u p p e r c r  cut  in  flow  Timber  be  of  thought  while  treatment  from  stock.  commodity  can  problem, the  planting  constrained  exceed  'within'  cash  for  that  choice  units,  policies,  requirements  requirement  the  treatment  f o r e s t planning  are  across  time,  constraints  constraints  the  or  a  action.  'bet ween are  u n i t over  r e s t r i c t i o n s on  category  constraints  interest  r e l a t i o n s h i p s between t h e  treatment  management The  some c o m m o d i t i e s o f  (production  unit,  structural of  cn  the  last  such  example, is  the  a  bound.  These  commodity  as  nursery typical  volume  i n some p a r t i c u l a r t i m e p e r i o d  lower  two  commodity a l l o c a t i o n  r e g u i r e m e n t s , and  RAM  as  units.  considerations  For  of  must  flow not  allocation  31  constraints  will  be r e p r e s e n t e d  for  a l l J constraint The  second  fluctuation action. treatment equation  category  a l l U treatment  of  a  unit  third  category  on  the d e c i s i o n  thinning. interval  restricts  the  i n r e s p o n s e t o a management usually but  embodied  will  in  the  be r e p r e s e n t e d i n  N  diameter  and Q management  of c o n s t r a i n t s  state  dimensions.  determines  actions  of a  the  elements  f e a s i b l e on t r e a t m e n t into  account  t h e amount o r i n t e n s i t y o f management  actions.  a restriction  variable after  B.C. F o r e s t  a  Service  must  take  on t h e i n t e n s i t y o f t h i n n i n g  defined  thinning,  (.8,.95). In g e n e r a l ,  represented  (3.1-6)  (v ,a ) = 0 qu u u  u a t time t . C o n s t r u c t i o n  average  be  are model,  /  units  example, c o n s i d e r  with  states  , t h e s e t o f a l l management  restrictions For  constraints  form as  h  The  of  simulation  i,  for  (3.1-5)  commodities.  relationships  unit  as e q u a l i t i e s ;  Dut  o f t h e management  These  form  l =0  g. (c. D  i n general  as  to  guidelines  the average  6  ratio  of  stand  diameter  before  restrict  these d e c i s i o n  a  t  to  restrictions  the will  by  & Provisional allowances for intensified stand management p r a c t i c e s f o r use i n a l l o w a b l e c u t c a l c u l a t i o n s , B.C.F.S., 1972.  32  a  for  A  Eqs,  the  an  Problem  intervals.  variables  the  meet t h e r e s t r i c t i o n s  management a c t i o n s  comprise  a  expressed  in  Formulation  timberlands  a**  optimum  planning  problem  r e l a t i o n developed = { a  and  | u<£E} such  u  D  h  a  qu  ut  approach  *  = R(c,v,a  can  i n section  now  be  3.1.  that  **  (3.2-1)  )  that  g . (c.  One  state  (3,1-7), then  R  is  and  T time  f  terms of the Find  and  solution, a .  General  The  units  decision  (3.1.5) -  feasible  (3.1-7)  ut  a l l 0 treatment If  3.2  £ ut ^  Dut  ) = o  (v ,a ) = u u  A  j  0  —  (3.2-2)  1,..., J  (3.2-3)  q = 1 , . . . ,Q u = 1 , . . . ,U  (3.2-4)  ut  to s o l v i n g  the  above  problem  is  to  separate  33  the  general  problem  of  allocating  into  the  commodities  management  seguences  seguences,  and  management referred To  feasible  smaller  values, be  The  MP 1,  problem  A will  of  through  possible  of c r e a t i n g  and  inner  problem  1  selecting management  the alternative  problems  will  l e t a be a c a n d i d a t e s e t o f management such  £: .  that  i t  contains  I t i s expected that s e t of  ( I f any management n o t be f i n i t e ) .  at  a will  a l l possible  action  least  generally  one be  management  c a n t a k e on c o n t i n u o u s  Then t h e commodity  subprcblem can  e x p r e s s e d as f o l l o w s :  13P1:  Find  a* C  a  such  that  z = R(c,a)  is  be  respectively.  than the complete A.  an ' o u t e r  interest  subset  outer  and MP2  problems;  of  a  'inner*  constructed solution  sequences,  frcm  sequences.  formulate  sequences,  much  an  t o a s MP1  two e a s i e r  (3.2-5)  a maximum, and t h a t  g.(c.) = 0  MP1  i s much  management limited  easier  seguences  j = 1,...,J  to solve to  to the r e s t r a i n e d  Z(a  *  be  than t h e g e n e r a l problem  considered  are  (3.2-6)  as the  pre-defined  and  s e t a . Of c o u r s e ,  ** ) < Z(a )  (3.2-7)  34  and  another  improved  of  MP2  problems  problem  difficult  the  to  provide  management s e g u e n c e a  o f f i n d i n g an o p t i m a l when  must be s o l v e d  to a .  management s e g u e n c e s  The not  series  treatment  units  are  *  i s  considered  separately.  MP2:  For each t r e a t m e n t u n i t management a c t i o n s ^  a  u, f i n d =  u  t h e sequence o f  '  ( a , , a „ , . . . , a u l  u2 '  )  m  such  uT  y = R (v , c , a ) U  is  maximized  XL  U  (3.2-8)  U  and t h a t  h  (v  uq  u  *  ,a  u  )  =  0  * 4-  A  However,  -  A  ut  MP1  u t  because  the  a r e not c o n s i d e r e d *  sequences feasible proper examined  1  i n MP1.  2  commodity  i n t h e MP2  *  a , a  U  =  1,...,U  q  =  1,...,Q  u  =  1,...,U  t  =  1 , . . . ,T  (3.2-9)  (3.2-10)  allocation constraints of  problem, a s e t  of  management  * ,  that  are optimal  The r e l a t i o n s h i p between  formulation  of  i n depth l a t e r  Direct  that;  optimization  MP2  as  in this of  after  a  f o r MP2,  MP1  may n o t be  and MP2,  subproblem  and  o f MP1, w i l l  the be  chapter.  the  general  impossible  except  simplifying  hypothetical  management s i t u a t i o n s . T h i s  problem  i s  assumptions thesis  will  usually or f o r  examine and  35  develop  solution  approach: problem  two  level  based  solution  of  on  a  decomposition  a commodity a l l o c a t i o n  and a s e t o f o p t i m a l management s e q u e n c e s u b p r c b l e m s .  the next with  the  strategies  section,  the f o r e s t  multistage analysis  structure  l a n d s p l a n n i n g problem  to display  and t o d e m o n s t r a t e  i t s underlying  i t s possibilities  is  In  examined  mathematical  for  mathematical  decomposition.  3.3 M u l t i s t a g e A n a l y s i s  It  w i l l be shown t o be a d v a n t a g e o u s t o f o r m u l a t e t h e f o r e s t  management multistage arranged  problem system so  which t h e n e x t sequential  as  a  consists  t h a t each one i n  problems,  of a  decision the  multistage  series  of  system.  decisions  A  a f f e c t s the circumstances  sequence  must  be  made.  stage,  and  thus  objective  of t h i s  type  of  affects  a l l those  formulation  one s t a g e a t a t i m e .  is  to  serial  that  In  t h e i n p u t t o any s t a g e i s t h e o u t p u t  previous  systems p a r t i a l l y ,  serial  is  under these of the  f o l l o w i n g . The optimize  large  36  3.3.1 ME1  As A 2 - p o i n t  The  MP1  sequences  decision *  commodities  Boundary V a l u e  '  a 2  problem  •••» u )  are a l l o c a t e d  manner. The f o r m u l a i o n  i s to select such  a  Problem  that  certain  constrained  a c r o s s t h e management u n i t  o f MP1  as a s e r i a l the  system  explicit  incorporation  of  allocation  must s t a r t w i t h  known commodity  w i t h t h e commodity  a s e t o f management  commodity  levels in feasible  i n optimal  requires  the  constraints.  The  l e v e l s and  must  end  region.  S t a g e : u, u=1,2, . « . , U At  stage  selected  Decision  u of the s e r i a l f o r treatment  variable:  a C  u  decision  selected  unit u  A  u  The  variable  from  s y s t e m , a management s e q u e n c e i s  is  the candidate  the  management  seguence a  set a . u  State  variables:  C  =  (C. , C.  u  l u  ,  2u  c  ) J u  (3.3-1)  The of  input total  state  C.,  initial and  at stage u i s the J dimensional  u  commodity  C. :u  The  C  = i  l e v e l s prior to scheduling  U-l  T  I  E  =  1  t  c. u t  j  usually  u.  (3.3-2)  l , . . . , j  D  value of the t o t a l  is  =  vector  commodity  state  z e r o f o r a commodity  that  is  known  i s t o be  37  produced  on  boundary  value  consumed  each  treatment  unit  has a p o s i t i v e  value  final  j  value o f the t o t a l  constrained  between  LB.  State  The  when a commodity i s  (e.g. f u n d i n g ) . c..=c°  The  (e.g. volume).  =  commodity  upper and l o w e r  <  ~  1  C. _  J  1  <'-> 3  state  is  3  3  usually  bounds.  <  (3.3-4) UB.  \  *  I  transition function:  T C. ]u  = W. (C. ,a ) = C. + :u ]u u ju  t  The  transition  =  i c. ]Ut  function  commodity  state  management  sequence a  (3.3-5)  1,...,J  j =  1  at  resulting  stage from  u computes t h e t o t a l  the  to treatment  application  unit  u. The  of  incidence  identity i s  C  ]U  Heturn  = c.  _.  (3.3-6)  t  ],u+l  function:  R  =  R  (  C  U '  C  U - 1  C  l '  a  i '  a  2 V  = E R n  u=l  The  stage  management The  total  return  from  seguence r e t u r n s return  u  (C , a ) u u  managing described  or objective  u  (  with in  function  a  u  section defined  3  '  is  3  "  7  )  the  3.1.4. over the  38  whcle  management u n i t  seguence As  the  returns  initial  total  commodity s t a t e  value  problem.  inversion,  selected total  and  the  function.  This  implies  output  = w.  the  reguire  final  decision  of the r o l e s of in  the  decision  transition  (c. ,a )  j u  = w!  j u  (3.3-8)  u  and C  u  , to  u  give  (c ,c )  311  u  (3.3-9)  and  i s a t w o - p o i n t boundary  variable  i n terms o f C  u  a  Eg.  exchange  state  ju  for a  management  that  c.  be s o l v e d  state  problems always  mathematical  variable  can  commodity  value  of a l l the  as e l e m e n t s o f a.  a r e f i x e d , MP2  Boundary  the  U i s t h e sum  u  (3.3-9)  u  shows what d e c i s i o n  i s needed  to transform  C  to C , u  The  computational  ignored  f o r now  The depend the  cn  decision function  and  (Eg.  with  function  inverse  the U output  be  3.4.1. can  C , the f i n a l l  using  can  be  made  state  C  to  , and u  the  state,  form.  First,  state  in  the  transition the f i n a l  0  objective  (3.3-7)).  R  the  by  decision  i s replaced  state  a p p r o a c h f o r MP1  in section  objective  decisions,  its  of t h i s  be e x a m i n e d  the i n i t i a l  remaining  function  Using  but w i l l  management u n i t only  U-1  feasibility  u  "  R ( c  u' u-l"""' i' i ' 2"" c  transition  c  function  a  :  a  , a  u-i  f C  u)  W , the s t a t e s u  ( 3  *  3  1  0  )  C ,C , ,..,C can 2 3 u  39  be  eliminated,  R  =  R ( w  u_i -" (  w  2  ( w  i  i' i  ( c  a  and t h e d e c i s i o n i n v e r s e provides  the f i n a l  R = =  Thus, to the  find  u  R  u  1  1  '•••'  of  a  u  _l  ' V  )  the  2  1  1  1  (3.3-11)  transition  2  2  u  1  function  u  1  (3.3-12)  sequences  conditions  state  W  )  u  the t w o - p o i n t boundary  value o p t i m i z a t i o n  (a  a ,  1 #  a^  2  )  problem i s such  that  a r e met,  3  < c.  <  DU -  transition  UB. j  j  =  functions,  (3.3-13)  l , . . . , j  expressed  as  equality  a r e met,  c. 3  manaqement  '  -w U  +  1  J  U  (c. , a ) 3u u  =  o  j = J  =  I , . . . , J 1  (3.3-14)  D  sequences are f e a s i b l e ,  a  t h e management  u = l,...,u  c A u^—  and  form  1  3 -  the  )  (C ,a ,a ,...,a _ ,'G )  LB.  constraints,  2  R(W .(W _ (...W (W (C ,a ),a ),...,a _ ),C  ]i  the  a  d e c i s i o n a^  t h e management  boundary  '  )  u  unit return  function  i s maximized:  (3.3-15)  R = Maximum a ,...,a 1  The  R(C  appropriately  in  2  u  sequential  schematically  (3. 3-16)  ,a ,a » ... ' ^ ^ ' ^  structure  Figure  3,  of  the  problem  with  the  stages  is  presented  represented  numbered r e c t a n g l e s , and t h e a r r o w s i n d i c a t i n g  i n p u t s and o u t p u t s  by the  to the various stages.  RU  U  V  F i g u r e 3.  MP1  as a 2 - p o i n t boundary v a l u e problem.  L  B  j  ^C  j  U  ^UB  j  41  3.3.2  MP2  For  As An I n i t i a l  each i n p u t  sequence This  a  means t h a t  inner  problem The  a^  =  must  a u T  some  management  such t h a t  )  t  c  D  MP2  management  finding a  y  a single  treatment  described  below:  optimizes  unit.  a  management  (3.3-7) i s m a x i m i z e d . stage  o f MP 1,  a management  to treatment  objective  described  that  MP1,  o f each  i s to find  U. The f o r m u l a t i o n easily  Eg.  state  applied  €  of  the  solved.  problem  unit  i s most  problem  o f each s t a g e  must be  decision  optimize  Problem  f o r each output  u l  system  state  be c r e a t e d  MP2  <a ,  Value  unit  defined of t h i s  after  sequence  u that  will  over  the  whole  problem  as a  serial  developing  an o b j e c t i v e  a  defined  The components o f t h e s e r i a l  simpler only  over  system a r e  S t a g e : t t=1, ... , T Each the  Decision  stage  t corresponds t o a d i s c r e t e time i n t e r v a l i n  time frame o f the system, d e s c r i b e d  variables: a .e Vi  ut  The  A  action  3.1.2.  • u t  decision variable  management  State  ^-  in section  i s the choice  t o be a p p l i e d  and  intensity  of  a  to u a t t.  variables: v ut  The  input  state  management s t a t e  of the s e r i a l described  v a l u e o f t h e management as  the current  i n section  state  management  system  (v  state  u l  at stage t i s the  3.1.3  . The  ) i s known and  of the treatment  initial defined unit.  42  v  in  this  free  , u l  =  formulation  as a c h o i c e  When MP2  state  unit,  commodity  only  not  as  the i n i t i a l  states  need  C. , and c. Jul  management  v° , and may  and  problem, n e i t h e r  State  the f i n a l  i s t o be o p t i m i z e d  tr€atment  states  (3. 3-17)  v ° u  state  be c h o s e n  with  respect  part nor  is left  optimally. to a single  o f a management  final  values  t o be c o n s t r a i n e d  of  unit the  and a r e c h o i c e  , respectivley.  JUT  transition function: v  The  ut  state  simulation  {  The  v  /  M  =  M  u  (p  ,v  u  u t  ,a  transition model Eg.  u , t  +  l '  C  u , t  relationship  +  l  }  i  u t  (3.3-18)  )  function  is  simply  the  stand  (3.1-1).  =  of  M  u  (  {  V  the  u t '  u t  C  }  '  output  a  u t  )  state  v  to the input  ut  state  v  a t any s t a g e  V  The  Return  state  ut  =  V  i s given  identity  (3.3-19) u , t + l  transition function  predicts  state  a t s t a g e t+1 as a f u n c t i o n  state  and management a c t i o n  function:  by t h e i n c i d e n c e  of  the treatment  the  at stage t .  treatment  unit unit  43 R  = R (v  u  u  T Z  = t  The is  =  ,v  ,,...,v ,,a ,,a  . — , a )  ul ul u2  u,T-l  uT  ^  •3-20)  3  R (v , a ) ut u t ut  1  stage  return  the a c t i o n  total of  uT  from s c h e d u l i n g  return  return  described  a management a c t i o n a^ in  section  f o r a management s e q u e n c e  a l l the action  returns  incurred  3.1.4.  B^fa^)  in  t  The  i s t h e sum  the  management  seguence. The can  of stages,  be c h a r a c t e r i z e d  value  state  sequential  structure  t o make t h e t o t a l  only  the  transition  one-state  is  known  described variable  and  fixed, i n each  above initial and  T  interval  frame.  exploited cn  and d e c i s i o n s  must be made on t h e management a c t i o n  the time The  states  as a T - d e c i s i o n ,  p r o b l e m . The i n i t i a l  decisions in  system  decisions  function  of  the  manaqement  serial  system  c a n be  sequence  return  depend  and t h e i n i t i a l  E g . (3.3-18)  state  to eliminate  v  states  by u s i n g t h e v  through  v uT  R  substituting  u  uT  ,v  u  unit  management a c t i o n s u, t h a t  ,v .,a ,,a „,...,a ul ul  function  u2  H  )  uT  value (a  ul  maximum r e t u r n  , a  a u2  (3.3-21)  ,  = R (M (...(M (v ,a ',P..),a _ ) , . . . , a _) ) u ut ul ul ul u u2 uT R (v ;a ,a ) u ul ul u2 uT  summary, t h e i n i t i a l  the  —  m  u,T-l  the t r a n s i t i o n R  In  = R (v  u  uT  (3. 3-22)  problem i s t o  find  ) f o r each treatment  o p t i m i z e t h e management s e g u e n c e  return  44  * R u  such  that  current  * * * = R (v ;a ,a ,...,a ) u u l u l u2 uT  the  initial  management  management s t a t e  state i s defined  of the treatment  (3. 3-24)  v  ul  t o be t h e  unit,  = °  v  the  (3.3-23)  ul  equality constraints  implied  by t h e t r a n s i t i o n  functions are  met,  v  and  that  - M ( v , a , p ) u ut ut u  u,t+l  t h e management  a  The  in  appropriately  actions  C A  are f e a s i b l e .  structare  Figure  4,  of with  numbered r e c t a n g l e s ,  and o u t p u t s t o t h e v a r i o u s  n -•— R  =Vl  R  u1  1  problem  the  stages  is  represented  stages.  ut  T  VI  MP2 as an i n i t i a l v a l u e  presented by  and t h e a r r o w s i n d i c a t i n g t h e  V  'u1  F i g u r e 4.  (3.3-26)  the  t L  (3. 3-25)  u = 1,...,U  ut  sequential  schematically  inputs  ut  t = 1, . . . ,T  =o  problem.  45  3.3.3  MF2  As A S u b p r o b l e m Of MP 1  Solution optimum  o f t h e MP2  management  initial  seguence,  value  a ,  problem  for  u  each  a  in isolation  feasible  MP1.  candidate  However  the e t h e r s .  set  f o r MP1,  feasibility  constraints commodity  from  can  only  with be  If  then  allocation constraints  accomplished  through  state  a t time  i n t e r v a l t , i n the f o l l o w i n g  State  variable: Let  Cj  Cj  after  scheduled,  the  *  JL  2  to by  the  is  y  optimal i n commodity  incorporating into  total  MP2.  the  This  commodity  is  input  manner.  u t  be t h e t o t a l  u t  consumed  time  representing  *  i t i s also  directly  unit  a , a , .. ., a  regard  guaranteed  provide the  treatment *  considered  will  amount  treatment  and t r e a t m e n t  o f commodity  u n i t s 1, 2,  unit  j  produced u-1  or  have been  u has been s c h e d u l e d  up  to  period t-1.  t - i C  State  j u t  =  C  j u  j = l,...,J  ^ f ^ j u k  (3.3-27)  transition function: C  jut  =  w. rc. . , a .) j u t j u t ut  = C.  + c.  jut  The  state  (3.3-28)  j u t  t r a n s i t i o n function  w .  simply  adds t h e amount  JU.  of  commodity  j produced  or  consumed  at  stage  t  as  a  46  function  of  the  management  action  a  , to the  running  total. For initial MP2  stages value  0-1  problem,  o f MP1,  C  At  Eg.  j u l  =  is total  u input state.  formulated commodity  By  t h e MP1  as state  an of  incidence  (3.3-6),  C  stage  u j  =  C  (3.3-29)  j,u-1,T  0 o f MP1,  the f i n a l t o t a l  output  MP2  with the i n i t i a l  d e f i n e d as t h e MP 1 s t a g e  identities  with  1,  MP2  i s a 2 p o i n t boundary  commodity  state  value  d e f i n e d by t h e MP2  problem U  stage  state:  ~  =  ~  The unchanged.  (3.3-30)  Q  j u  DUt  other  elements  The combined  schematically  i n Figure  of  MPT-MP2 5.  the  MP2  serial  decision system  problem is  remain  represented  47  c?  1  5,5  c r JO  a?  rr  n  >5  1  or  5-5  o (0  ra o  l  cr  U  o  (0  cr  ra  cr  CJ  ZdlAl  cu u Cn •H  48  3.3.4  The  Serial  The  Multistage  forest  lands  terms of the s e r i a l  Find  planning  multistage  problem  c a n now  U  I  in  such t h a t  T  E  i  , t=l  u=l  optimized,  be e x p r e s s e d  model:  t h e D x T s e t o f management a c t i o n s a  R=  is  Model  R  ut.  ( c ,v ut  ut  and t h e commodity  ,a )  (3.3-31)  ut  state transition  c o n s t r a i n t s are  met  C.  ,  ju,t+l  for  a l lj,u,t,  with  -  w. (C. ,a ) = j u t j u t ut  u # 0, and t  *  0  ,n  -3  so\  (3.3-32)  T.  The  initial  commodity  values  C , , = c° nil D  can  be  directly  conditions  substituted  are represented  g.(c.  The  (3. 3-33)  into  Eg.  (3.3-32) b u t t h e  final  explicitly:  ) = o  management s t a t e t r a n s i t i o n s  j =  must be  met  l , . . . , J  (3.3-34)  49  ~  V  u,t+l  for  can  ut  A  ut  4.)=  (3.3-35)  0  ut  management  = v°  into  Eg.  constraints  £2. A  A p p r o a c h e s To  By Dynamic  programming  e x p l o i t s the s e q u e n t i a l  system t o t r a n s f o r m  problem  i n t o a s e t of u x T o n e - d e c i s i o n  short the  the U x T - d e c i s i o n ,  o f d y n a m i c programming  (Nemhauser,  intuitive  (3.3-37)  Programming  serial  elsewhere  met:'  Optimization  Decomposition  The t h e o r y  must be  u = i,...,u t - 1 T  ut  Dynamic  (3.3-36)  (3.3-35).  t h e management a c t i o n  ut  state  u = 1,. .. ,U  u  ul  a  3.4.1  4.'  ( V  1. The i n i t i a l  be s u b s t i t u t e d  Finally,  3.4  ut  a l lu t, t £  v  4-  M  review  1966 will  structure J-state  of  a  decision  J - s t a t e problems. is  covered  ; W i l d e and B e i g h t l e r ,  extensively 1967)  but a  e m p h a s i z e some o f t h e a s s u m p t i o n s o f  technigue. A  dynamic  programming  decomposition  of  the  serial  50  multistage solution  model  space  problems  formulation  (u,t)  at that  (u,t)  values.  Such  as networks  combination.  s t a t e and  discrete  variable  w i t h t h e nodes a t e v e r y  F o r example,  o f t h e MP 1 s e r i a l  c  ,  multistage  the  discrete  problem  would  }  ut  i s the stage  stage.  material  A linear  and C  programming  to sequentially  MP1, t h e maximum  maximum  return  programming  states  are  a t each  evaluate  return  would  the s e r i a l  t h e u t maximum  function  return  R i s defined  At t h e f i n a l  selected  include  structure  a  to  of the  function.  as the stage 1  f , and i s c a l c u l a t e d  recursions.  arbitrarily  network  variables  node.  exploits  function,  dynamic  i s the value o f state  model o f t h e  balance c o n s t r a i n t  Dynamic  For  the continuous  nodes {  system  discrete  s t a t e and s t a g e  variable  where  with  replacing  c a n be r e p r e s e n t e d  possible  involve  involves  with  stage  fulfill  the usual  0, t h e o u t p u t the  boundary  conditions LB. < C. < UB. DDU D  For  every  decision  f e a s i b l e combination  inversion  appropriate treatment  form  decision unit  U.  of input  of the t r a n s i t i o n  and o u t p u t s t a t e s , t h e function  s e t (management seguence)  The  D  stage  maximum  returns  the  a^ t o be used on  return  involves  no  optimization  f (c ,c ) = U  last  u  u  stage  Rfw' (C ,c ) ,c ) u  Knowing  the  function  W, one c a n d e t e r m i n e  u  u  maximum the  (.3.4-1)  u  return (U-1,  and 0-2,  the ... ,  transition 2)  stage  51  returns  f  from  U  the following  (C ) = Maximum (R (C ,a ) + £ U  a  - _ *• u A u u C  initial  the  u  , (C ,a ) >  u+1  u  u  (3.4-2)  J  initial  s t a g e , t h e boundary  c o n d i t i o n s define the  state,  f (B) 1  = Maximum { R^ (B,a ) + f ( B , a ) } 2  a  The  u  u = U-l,...,2  for  At  recursion  i ^  A  (3.4-3)  2  l  process  i s repeated  f o r various feasible  values of the f i n a l  conditions,  as d i r e c t e d  by an optimum s e a r c h  technique.  Of c o u r s e , c o m p u t a t i o n solution  o f t h e above  o f t h e MP2 d e c i s i o n  When u = D, t h e boundary  C,  =  JU.T.  must  and  MP1  c a n be s o l v e d f  recursions  imply  s t a q e u.  condition  "1 — l r * f * f J  JU.  similar  At e a c h  a t each  C,  be met. The 0, T s t a g e  inversion,  problem  MP1  t o Eg.  staqe  r e t u r n i s computed  through  decision  value  problem  (3.4-1).  u = U, MP2 i s an  initial  directly,  . (C , v ) = Maximum {R (C , v , a . ) + f (C ^_,v ,a )} ut ut U t _ ut ut ut ut u,t+l ut ut ut a t. A ut ut t  =  T,T-1,...,2  u = U-l,...,1  (3.4-4)  52  by  means  of  the  transition  functions  H  and W u  sections  3.3.2 and 3.3.3.  fit t h e the  f  ul  initial  initial  (C  stage  t = 1 t h e boundary  conditions  describe  state.  ,V ) = Maximum {R u  ul  described i n  u  a  , . £ A ul ul  ul  (C  ,V  ul  ,a  ul  ) + f  ul  u2  (C . ,V . ,a ul  ul ul  )}  (3.4-5)  u = 2,3,...,U-1  when  f  l l  u = 1  (  B  ,  V  l l  )  =  M  a  x  i  l l ^  a  Although multistage the  u  m  A  {  ii  (  B  ' ii' n v  a  )  +  f  l2  t h e above  each  recursions  feasibility  value  of every  state  commodities  production  management  Goulding's management would  V  ll' ll a  )  (3.4-6)  }  theoretically  and  i s usually  management  the  than  2 o r 3,  be t h e s t a n d  The  in  time  Chapter  If  (J  =  20).  2)  was  list.  there volume  horizon of Similarly,  F o r example, i f used  f u n c t i o n , t h e management  age and a DBH  Hence  (J) a s  a planning  may be m u l t i d i m e n s i o n a l . in  program  constrained  periods.  per decade over  transition  problem,  i m p r a c t i c a l when t h e d i m e n s i o n s  constrained  (discussed  the  stored.  h a s as many e l e m e n t s  state  state  decision  results  a r e 20 e l e m e n t s i n C  model  optimize  variable C  i s constrained  20 d e c a d e s , t h e r e the  ,  s t a t e v a r i a b l e , a dynamic  the state variable are greater  are  B  o f the approach i s d o u b t f u l .  must be p e r f o r m e d  programming  commodity  (  model o f t h e t i m b e r l a n d s  optimization dynamic  R  l l  computational For  of  m  as  the  state v  53  The  replacement  decisions  with  problems.  of  discrete  The r e s o l u t i o n  increases  with  computational  the  effort  naturally variables of the  number also  of  transition  function  decision if  the  treatment  unit  problems state  transition  model, d e c i s i o n  even m e a n i n g f u l .  the  iterative  is  Decision  techniques  i n Appendix  and  computational  variable  formulation  classes,  but of c o u r s e  concern  the  transition  is  I f the  f o r the s t a t e . But  a  stand  or  may be i m p o s s i b l e o r  may  of approximating  function.  and o u t p u t  inversions  decision  solved  function  inversion  be c i r c u m v e n t e d Lagrange  I I , but t h e c o m p u t a t i o n a l  with  multipliers  effort  reguired  considerable. Most o f t h e c o m p u t a t i o n a l  programming MP1.  discrete  i n terms o f t h e i n p u t  not  described  causes  i s s i m p l e enough i t c a n be  state  states  increases.  management  variables  t h e management  also  of d i s c r e t e  However, t h e most s e r i o u s inversion  continuous  The  initial  approach ME2  value  are r e l a t e d  difficulties  programming, as w i l l  is  the  dynamic  t o t h e commodity c o n s t r a i n t s  problem, independent problem,  with  guite  be d e m o n s t r a t e d  o f MP 1 and f o r m u l a t e d tractable in later  by  sections.  of  as an  dynamic  54  3.4.2  The  Discrete  The  Kuhn  Optimum P r i n c i p l e  Tucker  theorems  state that  f o r a wide c l a s s  constrained  programming  problems, a Lagrangian  formed  involves  constraints  that  optimize.  The  no  lagrangian  whatever  values  (minimize)  the  function  value  Lagrangian will  i t  Lagrangian  formed  by  x nw  as  illuminate  the  value  cf  to  the  only  variables),  computational  deal  of  the  conditions  device,  the  a n a l y t i c power.  Here  under  which  occurs.  expression  adjoining objective  x T matrix of  inequality  a  o f f e r s a great to  equality or  f o r the  to  maximize objective  (minimize)  be  easier  original  (subject  conditions  decomposition  (3.3.4-4) t o t h e J  maximize  can  that:  variables  the  its  usefulness  approach  multistage  is  its  use  The  of  Lagrangian expression  Besides  i s generally  useful property the  to  will  non-negativity  we  the  of  subject  constraints, the  has  and  expression  of  f o r the  the  serial  constraints  function  multipliers,  Eq.  X_. . i i +  multistage Eqs.  (3.3.4-1) by and  model  (3.3.4-2) employing  a 0 x T matrix  a  ^  a  .  j L(X,a)  =  E ut  E  *  {R  -  EX.  UT - E X . 3=1  -  This X.  E  JUT D  E  a  analysis will  corresponding  to  g.ic. 3 JUT  ut  (v  u,t+l  [ ut  concentrate the  (v  ut  on  ,c  ut  ,a  the  constrained  ut  ))  Lagrange commodity  (3.4-7)  multipliers transition  55  functions, of  the  as  i t i s the  MP 1 and  MP2  transitions  commodities t h a t  problems.  will  be  Reference  emitted  are  to  to  the  the  common  elements  management  simplify  the  state  following  discussion. The  discrete  nature  of  the  optimum Lagrangian  order approximation) with  the  discrete  multistage  type of  exploits  expression  improvement  c h a g e s i n management The  technigue  to  optimum problem  principle  but  i s more  with  states.  most  of  information  selected  v a l u e s of  the  the  states,  on  guess  decisions,  set  resulting, improve  of and  the  the  then  much  less  trials  i f the  exploits  rather  first  function  how  by  deal  of  the  The  a  'best  the  states  so  p o l i c y improvement  or  with  them.  decisions  computer s t o r a g e a l t h o u g h g u e s s i s bad  to  dynamic  with  values  the  dynamic  continuous  a l l of  start  the  to a d j u s t This  with  serial  than  generated  than  generate  function.  initial  a  the  multidimensional,  maximum p r i n c i p l e  decide  objective  needs  objective  u s e d , i t would seem d e s i r a b l e  a l g o r i t h m s based 1  (with  efficient  systems  i s never  continuous  decisions.  for  programming  predict  i n the  programming Since  the  as  to  approach  i t may  require  function  is  many poorly  behaved. The  discrete  thoroughly elements optimum for  the  described of  the  (Fan  approach  principle  and  mcltistaged  initiate  matrix  of  the  Hang,  will  p r i n c i p l e i s a form o f  s o l u t i o n of To  optimum  be  1964), stated  s t a t i n g the  been and  widely only  h e r e . The  necessary  a  and few  discrete conditions  systems.  d e s c r i p t i o n of  Lagrange  has  multipliers  the is  algorithm, already  assume  that  known.  The  56  strategy problem  of  f o r each  Kuhn-Tucker problem  the  stage  optimum  such  that  approach  find  with  Hamiltonian  the  i s t o construct a  i t s solution  conditions of the o r i g i n a l  i s to  function,  discrete  stationary  problem.  points  of  satisfies  the  This equivalent a  Hamiltcnian  r e s p e c t t o t h e management d e c i s i o n s . The u t s t a g e  f o r the  serial  multistage  model  of f o r e s t  land  planning i s :  C . . ,a .) , E, .,E kR.m + 3I X. ]u,t+l.. w. jut(jut ut  ut  fi ^ =  The the  k=l_iH==l  partial  decision  derivative  o f the Hamiltonian  i s t h e Kuhn-Tucker  !!ut  =!V  ^ut When  3a  ut  +  dscisiion  EX. , ! ^ u t + 1  *  9a  the Hamiltcnians  ut  new s t a t e s v i a t h e  Lagrange process. process The  multipliers  =  then  respect  to  derivative:  «_£_  (3.4-9)  ' ut  6 a  by a d j u s t i n g t h e  d e c i s i o n s a r e used  transition  must  with  a r e made s t a t i o n a r y  management d e c i s i o n s , t h e i m p r o v e d the  (3.4-8)  functions. be  When t h e r e i s no improvement  obtained i n the  A to  t o compute  new  set of  i t e r a t e the  Hamiltonians,  the  terminates. problem  multipliers  will  of g e n e r a t i n g t h e a p p r o p r i a t e s e t s o f Lagrange now be examined. The L a g r a n g e a n e x p r e s s i o n  (3.4-7) i s u n c o n s t r a i n e d optimum a p p l y :  so n e c e s s a r y  f o r a l l u, t # UT  Eg.  c o n d i t i o n s f o r an i n t e r i o r  57  3R  3L 3C.  jut  9C  3R jut  3C  £  jut  J E i=l  recursive  3w  iu,t+l  3C  jut  iut  (3. 4-11)  3C.  'jut  relationship  j m u l t i p l i e r s to the  At t h e l a s t  (3. 4-10)  3w. iut  iu,t+l  1=1  jut  jut  commodity .  ut  ut  The  x. JUT  J  stage,  Eg.  unknown  the boundary  (3,4-11) r e l a t e s a l l t h e last  stage  v a l u e s impose  multiplier  the  condition  g. (C. ) = 0 j JUT y  and  X.  =  r  3  V  +  3 C -  Note  that  i  X.  i - l ' 1  T  i f  ,  U  T  +  1  t h e commodity  d  g  i  (3.4-12)  3C7  I transition function  w  j  is  iut  independent  o f commodity  3w  iut  j , then  = 0  i fi / j  (3.4-13)  3C ^ jut  as commodity  constraints  are usually  3w  ._ l u t  =  1  F u r t h e r m o r e , i f t h e commodity the  objective  function,  .  linear,  .  ifi = D  j i s not r e p r e s e n t e d  (  3, n -  1 4 )  explicitly in  58  3 R 3 C  If  =  jut  (3.4-15)  Q  .  t h e s e t h r e e a s s u m p t i o n s a r e used t o  all to  ut  t h e commodity the f i n a l  stage  X.  = X.  jut  j multipliers  = ... = X.  j,u,t+l  principle  consists  cf  the  najor  algorithm  generating  computational  strategies,  programming. decision  into  a solution  the  search  circumvent  i n value  and e g u a l  (3.4-16)  The  algorithm  problem cur  in  applying  serial  appropriate  we  two  algorithm.  *  commodity  inversion  decision  ,  to  the  to the f i n a l  requires  (3.4-16),  ju,T+l  corresponding  dynamic  are i d e n t i c a l  Eg.  multiplier  Consequently, optimum  simplify  a  multistage Lagrange  states.  model,  multipliers  Dpon  considering  e n c o u n t e r t h e same b l o c k as w i t h  point  boundary  value  p r o b l e m . MP1  to i n c o r p o r a t e the f i n a l  Again, as with  described  inversion,  discrete  dynamic  i n Appendix  conditions programming,  I I c a n be u s e d  but t h e c o m p u t a t i o n a l  effect  to is  considerable. The  discrete  optimum  multipliers  can  problem.  Lagrange  •glue'  The  which  separately, coordinating  be  used  approach to  with  link  multipliers  h o l d s the problem the  variables  d e m o n s t r a t e s how  Lagrange  the elements of a  decision  can  be  together.  Lagrange between  the  the  decision  state  variables  and  must  be  created.  i n value  a  as  serving  individual  e a c h a d j u s t m e n t o f an MP2  of  the  Each s t a g e i s o p t i m i z e d  multipliers  With  change  thought  optimizations.  variable,  new  set  as  of  the  commodity  multipliers  59  3.4.3 A p p r o x i m a t i o n With  The by at  linear  assuming least In  A Linear  programming  that  (LP) a p p r o a c h e s s e n t i a l l y  a c a n d i d a t e s e t , a, e x i s t s  contains  a l l t h e management  forming the l i n e a r  management  Model  a  sequence  u  coefficients  that  constraint  equation  £L  model, a  #  represent  sequence  t o one a c r e  general  commodity  of  and i s c o m p l e t e , o r  one  associates  of land.  application  Consider  constraint  with  every  vector of input-output  the c o n t r i b u t i o n s the  MP 1  seguences of i n t e r e s t .  column  a  selves  a  t o each of  linear  equation  the  commodity management  form  (introduced  of  the  i n section  3.1.4) U  T  v  y c.  g. =  :  The  I I  loss  treatment  of  unit  = C.  JUT  j u t  a s s u m p t i o n o f an  without  = C.  equality  (3.4-17)  j  constraint  generality.  L e t -x  simplifies  notation  be t h e number o f a c r e s o f  u managed by t h e a l t e r n a t i v e  management  sequence  (k)  a  . Then  Eq. (3.4-17)  U  g  j  "  3c.  T  *  can be r e w r i t t e n a s  I  X  u k  =  j  c  (3.4-18)  uk  Associated  is  with  t h e M a l t e r n a t i v e management s e q u e n c e f o r u 3C_ 3C. the vector of d e r i v a t i v e s . For e x a m p l e , — 3 — might be uk 3x , 3  x  uk  the  volume/acre  third  p r o d u c e d by c l e a r c u t t i n g  treatment unit  u i n the  decade. As w i t h dynamic  approach  to  programming,  optimization  will  a short  description  of the  LP  h e l p t o m o t i v a t e the development  60  of  a more s u i t a b l e o p t i m i z a t i o n In  terms o f 8P1, t h e  treatment  unit  Adjustment First, applying  among  of x  there a  LP  algorithm approach  allocates  alternative  a  direct  3.4,4. acres  management  a f f e c t s the o b j e c t i v e is  i n section  function  management  of  a  sequences.  B i n two  sequence  ways.  return  from  t o u, 8R  ax UK  Indirect  e f f e c t s cn R come f r o m i m p r o v e m e n t s  of  the  u n i t as a whole, t h r o u g h a l l o w i n g  the  constrained  commodity  c o m m o d i t i e s . The  i s computed  represented  implicit  a t each i t e r a t i o n  in  the  management  more e f f i c i e n t value  of a  use o f  constrained  o f t h e LP p r o c e s s and i s  as a d e r i v a t i v e , 6R 6  This  price  is  1967),  change  the  perturbations infeasible commodity  x  meaning  derivative  that  objective  the  function  i n t h e commodity  went  slack  (after  function resulting  variables.  would be i f t h e amounts  and  i s the rate of from  feasible  An example of  c f an  'excess' of a  effect  of  a  perturbation  of  +  9R_ 8  This  expression  as  the  X  u k  z *  the  uk  ^ 8  x  -SB. u k  6  C  (3.4-18)  j  f o r the decision  unconstrained  is  x  derivative:  =  Wilde  negative.  total  constrained  « uk  constrained  perturbation  The  6R_  C  a  Beightler, of  - J  d e r i v a t i v e can  management  sequence  be  interpreted  return  with  a  61  connection The for  f o r keeping  IP s o l u t i o n  t h e commodity c o n s t r a i n t s t i g h t .  algorithm  variables x  decisions  iterativsly  •,  and  computes  new  values  the associated constrained  uk  derivatives,  3R 7— 3x  V  -  JS. —  such  x  t h a t complementary  =-o =  , uk  3C. j  uk  1  3R  —  v  When n o n - p o s i t i v i t y  ||-  C. j J  3 ~  X / • • • / IS.  i  slackness i s maintained:  '' * •'  J  o f the constrained  derivatives  |*9C . D  the  optimal  function  Host  have been  to  o f Eg.  (3.4-17).  eguations,  the  handle  the  planning. codes  very  problem  a selection  problem  easily  of post  divisibility the  of the  constrained  expressed  LP has c l e a r  are  associated  algorithms,  optimal  in  advantages  t h e whole management  size  solution  objective  c o n s t r a i n t s i n the f o r e s t  l a r g e g e n e r a l i z e d LP c o d e s large  (3, 4-21)  of the  allocate  are  Consequently  In a d d i t i o n t o  offer  and  commodity  commodities are c o n s t r a i n e d across Furthermore,  J  the l i n e a r i t y  efficiently  of  (3. 4-20)  1,...,K  obtained.  exploits  management p l a n n i n g  form  =  j = 1  0  constraint  units,  commodities. land  -  programming  and  treatment  k  values  Linear  = i,...,u  u 0  uk  i s obtained,  procedures  when  unit.  available with  these  the  to  forest  computer  to a s s i s t i n  62  analyses of the planning  planning  models,  (Clutter  model.  Timber  RAM  The  most  (Navon,  e t a l , 1S69) c a p i t a l i z e on  successful  197 0)  these  and  forest  Maxmillion  features  of  linear  programming. The  disadvantages  that  the l i n e a r  real  planning  Although linear is  model i s a l e s s problem,  t h e commodity  eguations,  much l e s s  priori.  planning large  of  the  approximation  serial are  linear the  sequences necessary i f  a  of  multistage  readily  of  the  model.  expressed  28,000 a l t e r n a t i v e  management  later  Usually  sections.  solve  set  as  function  sequences  treatment  sequences  only  a  sequences are e x p l i c i t l y  of  MP2  created  for a due t o  unit  will  small  and  alternative  was  be i m p o s s i b l e  A  MP1  t h e optimum, i s known  variable  would  variables,  to  complete  management  problem, s o l u t i o n  number  model  t o compute  decision  a l l feasible  management  the  constraints  assumes t h a t  Yet,  represent  than  the f a c t  satisfactory.  simultaneously management  accurate  from  the l i m i t a t i o n s o f a l i n e a r o b j e c t i v e  Construction  a  o f t h e LP a p p r o a c h d e r i v e  with  be  realistic the  very  more  than  described  number  represented  to  of  in  in  possible  the  linear  model. In nature  summary,  the  of the c o n s t r a i n t  solution  to  reguirinq  the candidate  priori  MP1,  and i n c l u d e d  The  LP  approach equation  MP2  capitalizes and  problem  management  i n the l i n e a r  provides is  avoided  sequences  model.  to  on t h e l i n e a r an  efficient  completely be  known  by a  63  3.4.4  Dantzig-Wolfe  In  sections  problem  3.4.1  and  serial  through  multistage  problems  algorithms  structure.  (HF1), i n which t h e  computationally  »€  have seen  space.  improvement  The in  requires the  allocation  to optimize  optimum  boundary  space  and  but,  values of  forest  decision  a  usual  the  an  continuous,  with  c f the  dynamic  transition  commodity  reasonably  inversion  dynamic  provides  as  the  embedded,  multidimensional  for  inversion  function i s  simulator,  v i a the  efficiency  decision  are  approach  Where t h e s t a t e t r a n s i t i o n lands  units i s best  However, t h e commodity  to a continuous  discrete  f u n c t i o n s t o handle  exploit  MP2  problem's  computational  it  that  t h a t the  the  m u l t i d i m e n s i o n a l commodity s t a t e programming,  treatment  s c h e d u l i n g problems  infeasible  programming r e c u r s i o n s , due state  3.4.2  s c h e d u l i n g management a c t i o n s on  accomplished  is  Decomposition  states.  realistic  is unlikely  to  be  feasible. Conversely, solution  of  approximation  the  o f MP1  commodity efficient.  staged  problem  commodity and The  of t h e  constraints.  LP  commodity possible  approach  assumes  allocation. management  to e x p l o i t  If  model with  linear  be  is  set  serial the  staged  directly.  selected contains  i s s o l v e d through  and  linear  cf  time  handled  set  candidate  s e g u e n c e , MP2  form  the n a t u r a l l y  candidate  the  problem  the  problem cannot a  linear  T h i s approach i g n o r e s the  Consequently,  h i g h l y n o n - l i n e a r KP2  a  allocation  programming, i s v e r y nature  with  for every  exhaustive  enumeration. ft s y n t h e s i s  of these  two  approaches  is  suggested  by  the  64  role  of  the  solution  strategy.  variables  u"  key  combining  stage  of  the  commodity  constrained to  condition  X  =  ££_  more  intuitive  instantaneous  them  problem other  can  MP2  approaches  the  objective  be be  problems  lies  .This  t + i  to  in  the  with  the  multiplier i s  function  B  with  constraint.  (3.4-22) 6g.  jut  value to the c  coordinating  multipliers associated  i  interpretation  r i g h t hand s i d e ,  MP2  optimum  =  6c. 1  as  to the  two  state *  final  discrete  allowing  each  'glued*  Lagrange  d e r i v a t i v e of  the  stages,  the  output  the  serve  extension,  The  jU,T+l  the  T  Lagrange m u l t i p l i e r s .  respect  A  x  the  interpretation  the  the  i n d i v i d u a l l y , while  to  in  multipliers  i n d i v i d u a l l y . By  optimized  last  multipliers  The  between  optimized  with  Lagrange  of  objective  the  multiplier  function  of  is  another  the  unit  of  . JUT  The  LP  s o l u t i o n of  t h e s e commodity is  the  prices  optimal i s to  are  constrained  respect  to  The multistage  the  1  :  An  automatically.  efficient  commodity  model as  the  following  initial  problem.  of  Although  commodities,  commodity  derivates  identification  s u g g e s t s the  l i n e a r approximation of  allocation  problem the  set  the  of  the  prices.  the the The  objective  MP1  provides  MP 1  problem  dual  of  dual  this  variables  function  B  with  constraint. of dual  the  Lagrange  variable  of  multiplier the  linear  of  the model  algorithm:  feasible solution a  f  u  i s provided  f o r each  MP2  65  2 :  The LP  3  :  MF2  d e c i s i o n s a r e imposed  operations  Using  the i d e n t i t y  multipliers 4 :  The  either If  a r e computed  with  stage  accomplished to  repeated 6 :  from  step  forest  the optimal  lands  The  Hamiltonian  For a problem  the  set,  in  and  decisions,  step  4  was  seguences  are  the  Hamiltonians  process i s  in  step  4  was  terminates.  management cf  s e g u e n c e s i n t h e LP b a s i s  management  actions  f o r the  problem. is  a  construct  o f the  closely  constrained  step  4  that optimizes  MF2, and f u l f i l l s  non-positivity i s optimized.  be  decision derivative  prices, the  (maximization) This i s  related  decision  would  d u a l commodity  i f MP1  be w r i t t e n and  management  any s e t o f MP1  derivative  Lagrange  (3.4-11).  management  maximizes t h e d e c i s i o n d e r i v a t i v e  of  problem.  then  Hamiltonians  of the  approach at  condition  MP1  can  the  ut  v i a Eg.  but e x p l o i t s t h e m u l t i s t a g e n a t u r e  A more g e n e r a l that  stage  2 .  schedule  Kuhn-Tucker d e f i n i t i o n  problem  the 4,  the  planning  (3.4-9),  to  the process  termination,  comprise  (3.4-8)  candidate  I f t h e improvement  At  the  in  LP  6R  o r a l l a t once.  i n step the  the  recursively  respect  by s t a g e  insignificant,  Eg.  Eg.  improvement  added  (3.4-22),  model c f MP 1.  variables T — , i  a vector of dual  Eg.  Hamiltonians  optimized  5 :  provide  on a l i n e a r  to the  derivative  of t h e problem. tc  construct  a  directly. we can c o n s t r u c t MP1 of  Kuhn-Tucker the  accomplished  decision by  using  as t h e o b j e c t i v e f u n c t i o n o f t h e MP2  66  For  each treatment  unit  u, f i n d  t h e management s e q u e n c e a u  =  {  a  u l '  a  u 2 '  a  6x  3  x  uk  is  u T  J  £  fl  , uk  T'  j J  a maximum, and t h a t  S  U  C  h  SC. 3  t  h  a  t  (J.4-2J) uk  the  management  state  transitions  Eg.  (3. 3. 4-4) a r e met v  „ f x i u,t+l  The  -  M  (v  u t  decision  ,c  ,a  u t  u t  )  =  derivative  above. T e r m i n a t i o n of  u t  0  t  form  of the process  =  1,...,T  o f MP2 occurs  replaces steps when t h e maximum  the o b j e c t i v e f u n c t i o n i s zero or negative.  |f  1  3 and 4 value  When  (3.4-24)  0  uk  for  a l l u,  decision is  t h e Kuhn-Tucker c o n d i t i o n o f n o n - p o s i t i v i t y  derivative  maintained  i s met f o r MP1. As  complementary  of the  slackness  f o r HP 1 by t h e LP a l g o r i t h m and c o n v e x i t y c a n be  assumed, MP 1 i s o p t i m a l . This technique new  decision  Wolfe-Bantzig In  o f using the dual v a r i a b l e s  vector  at  each  decomposition  t h e next  subproblem  will  subproblems  will  chapter,  be p r e s e n t e d  MP1, v i a B a n t z i g - W o l f e  alternative  a Timber  decomposition.  a  1961).  forms  and o p t i m i z e d .  with  construct  o f an LP i s known a s  ( B a n t z i g and W o l f e ,  some  be combined  iteration  to  of  In chapter  RAM  the  MP2  5 these  formulation  of  67  4*  Qptifizatipn  The optimal  Of  The  Sufcprcblem  mathematical  structure  seguence of  management a c t i o n s  has  been  examined i n s e c t i o n s  land  u n i t c o n s i s t s of  techniques The  of  techniques stand  of  that  models,  can  management t o o l s . for  developed. efficient  4.1  The  Direct  methods  or  be  well  earned  model c f  used  the the  directly.  optimization  be  stand  behaved  in  the  acceptance  as  described.  First,  a  will  be  models  uses t h e  directly  valid  model embedded  in  an  formulation.  too  to o p t i m i z a t i o n ;  i n order  indirect  unit  functions,  describe  f o r the  objective  complicated 3.3.1  and  to 3.3.2,  approximation  methods. A p p r o x i m a t i o n t e c h n i g u e s e v a l u a t e points  land  the  to c u r r e n t l y a v a i l a b l e f o r e s t  will  a n a l y t i c expression  methods c f s e c t i o n s  approaches  could  to  stand  programming  I f the  analytic  generally have  s e c o n d method  unavailable  indirect  but  forest  3.4.  3.3.2  of f i n d i n g  Optimization  Shen an either  not  optimizing  dynamic  and  is  applied  are  Two  and  problem  for a  behaved,  chapter be  sense,  3.3  3.3.1  this  which  mathematical  technigue  well  sections  objective  of t h e  to approximate  methods.  (1972) f i t t e d  the  As  a  response  i t by  an  of  is  m a n i p u l a t e by  the  there  are  t e c h n i g u e s and the  two  direct  o b j e c t i v e a t many  expression  somewhat s i m p l i s t i c surface  function  amenable  example,  volume y i e l d  to  Goulding  over a  range  68  of  values  at  o f d e c i s i o n v a r i a b l e s , s u c h as i n i t i a l  harvest.  response could  As  surface  be s o l v e d Direct  stepwise direct  f  was  methods  toward  points  considered seeking  the  and  at  optimum  successive  a climbing  technigues  A general  climbing  methods  past  the  problem  t o stand  treatment  proceed  improvements.  information  procedure.  The  applicability  simulation  including direct  climbing,  Any  to generate  models w i l l  o f a wider c l a s s of  techniques  and  c a n be f u r t h e r  pattern  searches.  back t o Cauchy, u s e s t h e p l a n e  be  optimum  c a n be f o u n d i n  subdivided  The g r a d i e n t  tangent  to  the  into method,  response  t o i n d i c a t e t h e d i r e c t i o n o f improvement o f a p o l i c y . At new  steepest  point  ascent.  by  a  on  the  response  the l i n e  maximizing classical  is size  technigue  The  latter  of steepest this  ascent  expression  and t h e p r o c e s s  most e f f e c t i v e properties.  parallel  t a n g e n t s method  improved  the  along  gradient the  along  the g r a d i e n t  or,  when  for  the  of  direction i s  possible,  direct  s u b s t i t u t i n g the  eguation  length  At t h e new  is  is  line  i n t o the o b j e c t i v e f u n c t i o n ,  gradient  following  surface  involves  calculus techniques.  evaluated The  The s t e p  search  differentiation.  is  'optimization'  a r b i t r a r y p o i n t and  using  a p p r o x i m a t e d , and t h e p o l i c y  of  dimensional,  (1964).  surface  found  was two  the  an  by  scheme  techniques,  gradient  each  start  i s called  here.  Direct  going  plotted,  space  and age  by i n s p e c t i o n .  dirjc.t, c l i m b i n g  Wilde  decision  optimization  better o  the  density  of the l i n e  point,  a new  and using  gradient  repeated. technigues  In a d d i t i o n , two  have e f f i c i e n t  important  o f Shah, B u e h l e r  ridge  technigues,  and Kempthcrne  the  (1961)  69  and  the  deflected gradient  (1963)  have  deflected  and  Beightler,  derivative decision  the  requires  that  the  a d v a n t a q e of saddle While  the  above  objective  function  efficiency  recommended  the  they  be  optimization  s p h e r i c a l as  well  estimated  or  parallel  they  estimated. inherently  are  much  efficient  the  less  ideal.  Buehler  et  c f the  gradient  (1961) r e p o r t  have g r e a t l y  pattern  search  objective  the  a low  be  are  representation. approach  surface  of  was the  improve (1961)  choice  the  a in and  non-guadratic s e n s i t i v e to  Consequently,  f o r e s t stand  by  Buehler et a l .  cn  tc  of  expansion  accelerated  generally  chosen  where  function  Taylor  the  the  v a r i a b l e s , make  objective  performance still  To  al.  symmetric  order  on  as  transformed  developed since  improved  methods  s c a l i n g and  by  optimum. A l t h o u g h  algorithms  gradient  possible  by  efficient  quadratic  problems  represent  approximated  neighborhood  problems of  the  convergence,  supplied that  first to  gradient  remove i n t e r a c t i o n between i n d e p e n d e n t  surfaces,  or  techniques  optimization,  function  complicated  the  (Wilde  the  respect  form  The  guadratic  choice  quadratic  methods i s  become  measurement, and  robust  derivatives  d e p a r t s from  cf  that  c o n t o u r s as  deflected  best  with  quasi  gradient  s c a l e s of  the  of  procedure r e q u i r e s  Similarly,  Powell  points.  surfaces,  possible:  the  and  objectives.  property  analytical  with  a l l qradient  guadratic  the  1964).  first  quadratic  function  in  search,  Fletcher  considered  objective either  cf  the  However, t h e  (Powell,  (partan)  cn has  i s generally  variables,  tangent  avoid  approach  1967).  of  numerically  One  behaviour  qradient  convergence and  ideal  procedure  the  optimize model.  more the  70  Pattern various  stages  Conseguently the  searches  required. ridges  of  surface.  Hext and  idea  from  for  a system  space,  triangle  and  techniques,  to the neiqhborhocd  Himsworth  adaptive  (N=2)  and t h e r e g u l a r  causing  down l o n g  direction  on  contracting  find  o f t h e optimum  using  inclined  a t an o p t i m a .  Create  i s representative section.  (1962) i n t r o d u c e d a s i m p l e b u t through  forming  e v a l u a t i n g the a simplex  new  tetrahedron  i n the  simplices  of the remaining  generalization  t o adapt  encountering  maximization  are not  by  points.  o f the e q u i l a t e r a l (N=3).  Nelder  and  the a l g o r i t h m f o r o p t i m i z a t i o n a p p l i c a t i o n s  the simplex  elongating  in  searches  at a set of points forming  i s the N-dimensional  at  optimum.  derivatives  i n the next  control  continually  the  irregularities  pattern  one p o i n t i n t h e h y p e r p l a n e  (1965) i m p r o v e d  :  second  Algorithm  A simplex  1  or  gathered  for  to local  The S e g u e n t i a l S i m p l e x  reflecting  for  First  search  in detail  factor  by  the  i s presented  ingenious  Mead  information  methods. A s e a r c h a l g o r i t h m t h a t  Spendley,  output  them  use  sensitive  gradient  and f e l l o w  the approach  4.1.1  are l e s s  As w i t h  acceleration  to  throughout  they  objective  attempt  a  itself  valleys valley  A brief  to the l o c a l (or  landscape,  ridges),  changing  ( r i d g e ) a t an a n g l e , and  description  of  the  algorithm  fellows:  the v e r t i c e s  the o b j e c t i v e  of the i n i t i a l  function  a t n+1  simplex  by e v a l u a t i n g  p o i n t s i n the space  of  the  71  n independent 2  :  The v e r t e x (Xmin)  point  a t which t h e f u n c t i o n  is  centroid  determined,  (Xnew)  where a >1  by t h e  step 4 :  point  step  point  an e x p a n s i o n  proves s t i l l replace  defined  u(Xnew)  +  point  Xnew  of  maximum, go t o  t o be t h e worst  step  point,  Xmin by Xnew and go t o  go t o s t e p  by t h e  coefficient.  3 Xmin  than  return  simplex,  by t h e  to step  before  best  i n the simplex,  relation  If  coefficient.  If  Xmin i n t h e s i m p l e x Xcon  does  has f a i l e d  a r e moved  not  Xcon  is  w i t h Xcon  improve  the  and a l l t h e v e r t i c e s  closer to  the  best  point  relation  (X + Xmax)/2  going  reflection,  2.  the c o n t r a c t i o n  (X ) o f t h e s i m p l e x  X =  Xmin  the  + ( 1 - 3 )X  Xmin, r e p l a c e  (Xmax) by t h e  Select  2.  where 0<3 <1 i s t h e c o n t r a c t i o n better  relation  and Xexp t o r e p l a c e  Attempt a c o n t r a c t i o n  represented  trial  (1- u> ) X  i s the expansion  Xcon =  A  a  coefficient.  p r o v e s t o be a new  where u >1  and  give  2.  Attempt  and  (X) t o  relation  5. O t h e r w i s e ,  Xexp =  5 :  vertices  i s the r e f l e c t i o n  H. I f t h e t r i a l to  value  (1 + a ) - x - a (Xmin)  I f the t r i a l  go  takes the smallest  and i s o v e r - r e f l e c t e d t h r o u g h t h e  of the remaining  Xnew =  3 :  variables.  t o step  expansion  i n Figure  6.  2. and  contraction  operation  are  contraction  Figure  6.  Elements and  of  a  s i m p l i c i a l  c o n t r a c t i o n .  search:  r e l e c t i o n ,  expansion  73  The  sequential  optimization to  a  algorithm  number  d o e s not  of  reguire  function  simplex  m o d e l s , as  first  or  any  Information  function size  i s coded  varies  ridges,  and  compact, function  (model)  The w i t h few were  be  in  the  original  in  the  objective  the  landscape,  as  as  it  The 2,  1/2)  p a p e r . The function  as  i t e r a t i o n s s p e c i f i e d by  reached.  The  procedure  a t any  handle If or  terminal  deviation  'explicit'  a simplex  of  or  can  simple  upper and  projects  c o n s t r a i n t s , the  feasible  region  by  objective The  step along  computationally one  was  by  additional  implemented , u  or  ,  N e l d e r and  iterating  .001,  i f the  when t h e  OPTIMIZE command  an  original  improved  c o e f f i c i e n t s (a  interrupt  generating  from t h e  operation  more e x p l i c i t  border  by  user  the  an  search  iteration.  above,  i s l e s s than  a  decision  the  only  per  stops  is  a pattern  is  recommended  routine  only  accelerating  reguires  adaptation  search  simulation  simplex.  method  (simulation)  due  the  of  adapting  f i n d i n g and  number o f  One  the  models  This  generate  landscape  i t i s presented  (1,  time  model. As to  the  performing  formulate  n e a r o p t i m a . The  modifications.  chosen to  simplest  evaluation  algorithm,  to  as  simplex  (simulations).  shape o f  economical  f o r e s t stand  suitable for optimizing  about  shrinking  chosen  derivatives,  information  i n the  with the  and  second  the  p r o c e d u r e , i t uses past policy.  with  impossible  but  was  c h a r a c t e r i s t i c s . The  algorithm  i t i s usually of  tested  evaluations  f o r an  derivatives  be  favorable  (model)  prereguisite  to  method  the  3 ) Mead  change maximum  has  been  optimization  ATTENTION i n t e r r u p t .  algorithm lower  a trial trial  bound  point point  adjusting  i s the  the  to  constraints.  that is  ability  violates  moved  bounded  to  one the  variable.  74  •Implicit' decision  c o n s t r a i n t s , which a r e f u n c t i o n s v a r i a b l e , could  recommended  by Box  of  more  be accommodated by a s i m p l e x  (1965) , t h a t  would  move a t r i a l  t o w a r d s t h e c e n t r c i d i f i t v i o l a t e d an i m p l i c i t facility  was n o t i n c l u d e d  Modifications models. is  statistical  test  implemented described  i n Appendix  allows  simulation  existing  policy  This  criterion  but the  of  value  using  points  procedure  the a b i l i t y  a had  was n o t  algorithm  is  was p r o v i d e d  to  evaluations  at  i s c a l c u l a t e d as an  the i n t e r a c t i v e nature  specify  algorithms  7) was d e s i g n e d that  the  f o r e s t land  Supervisor  optimization  of providing  optimization  expected  to  This  stochastic  was d e v e l o p e d  repetitions  with  constraint.  more  accuracy  of the as  the  with  the  p o l i c y i s approached.  objective  (Figure  along  the user  The O p t i m i z a t i o n  A  I I I . Instead,  halfway  routine.  two  subroutine,  and t h e o b j e c t i v e  facility,  point  accommodate  objective values.  one  operation  o f the performance  whether  of independent  point  This  supervisor,  4.1.2  to  a procedure  decide  i n the optimizing  a number  average.  to  different  policy  optimal  considered  over i t s range,  significantly  each  i n the o p t i m i z i n g  Assuming t h a t t h e v a r i a n c e  homogenous  specify  were  than  Program  program  a supervisor to various to  be  was  written  f o r applying simulation  highly  nonlinear  and  management  models  direct  models. The s y s t e m  interactive  discontinuous would  climbing  as  i t  was  nature of the  reguire  frequent  75  Commands READ DISPLAY EDIT SET SIMULATE  Conversa-  FIX  \  tional  FREE  y  Terminal/  OPTIMIZE  User  Session  hardcopy  SUBROUTINE EOF STOP  Optimization  Supervisor  opttipiize  Sequential Simplex Algorithm  Goulding"!  Figure  7.  and  reports  WRITE  Meyers  >  Kilkki  J  The Simulation  Stand  Model  Library  Optimization  (SIMOPT)  supervisor  system.  7  intervention  for  human  decision  making  optimization  a n a l y s i s . The s u p e r v i s o r  is  dependent  heavily  character  handling.  IV,  The s u p e r v i s o r  are  briefly  on  MTS  The  utilities  7  the course  i s written  s u p e r v i s o r code  currently recognizes  described  in  o f an  i n FORTRAN  f o r input/output i s listed  twelve  6  but and  i n Appendix  commands,  which  to lead a  policy  below.  READ The  READ  with  a specified  unit. the file  command  The f i l e  program  and  optimization completely  identifier  label  from  POLICY i s a s s i g n e d  i s first  specifies  variables,  causes the supervisor  a  given  to a logical  s t a r t e d . Each p o l i c y simulation  additional procedure.  described  a  i n the  u n i t when  the  POLICY  model, i t s p a r a m e t e r s and  information  The  on  logical  POLICY  test  needed file  problems  f o r the  will of  be more a  later  section.  DISPLAY  The  DISPLAY command c a u s e s t h e c u r r e n t  and v a r i a b l e s t o be d i s p l a y e d internal  integer  with  their  parameters  corresponding  identifiers.  A l l d e v e l o p m e n t work was done on an IBM u n d e r t h e M i c h i g a n T e r m i n a l System (MTS). 7  policy  370-168  operating  77  EDIT The  EDIT  command  editor  system  policy  file.  available.  c a u s e s t r a n s f e r o f c o n t r o l t o t h e MTS  and i n i t i a t e s A l l the  the e d i t i n g  of  the  current  f a c i l i t i e s o f t h e MTS e d i t o r a r e  At t h e end o f t h e e d i t ,  the s u p e r v i s o r  restarted  without  re-initializing  the system.  The  command  allows  to assign  can  be  SET SET  current  the user  values  to the  p o l i c y v a r i a b l e s or parameters.  SIMULATE  The  SIMULATE command  current  policy  execution, terminal printing.  to  tabular or  causes the supervisor the appropriate results  directed  to  may a  average r e t u r n s  pass  simulation  be  file  the  the  model. On  displayed  scratch  A r e p e t i t i o n f a c t o r allows  to  at  for  the later  calculation  of  f o r s t o c h a s t i c models.  FIX The  .FIX  value  command  throughout  causes  a v a r i a b l e to hold  the o p t i m i z a t i o n  i t s current  process.  FREE The  FREE command  optimium  value  releases  during  a  variable  the optimization  to  take  process.  cn i t s  78  OPTIMIZE  The  OPTIMIZE command c a u s e s t h e t r a n s f e r o f c o n t r o l t o an  optimization iterations. free user of  algorithm The v a l u e s  variables  are  can i n t e r v e n e the  region  of  interest.  models.  improved  specified  o f the o b j e c t i v e  on  number  f u n c t i o n and  at each i t e r a t i o n  i f the algoritm  based  at each  a  displayed  optimization The  for  stalls  values  o r wanders  policy replaces  for  the  and t h e  A repetition factor  average  of  out  allows  stochastic  the current  policy  iteration.  SUBROUTINE  The  SUBROUTINE command  control  t o a user  causes  supplied  the  supervisor  to  pass  subroutine.  WRITE The  WRITE  command c a u s e s t h e c u r r e n t  on a s p e c i f i e d  logical  unit. This  an o p t i m i z e d  policy f o r further  An  file  p o l i c y t o be s a v e d  allows  the user  t o save  processing.  EOF end  of  supervisor the  input  operating  signal prompt  generated  will  system.  cause  Under MTS  c a n be s a v e d on permanent  files,  reports  line  copied  to  the  in  response  to  a  t r a n s f e r of c o n t r o l to , policy scratch  and  simulation  files  program  p r i n t e r . The s u p e r v i s o r i s  79  re-entered  without  re-initializing.  STOP The to  STOP command c a u s e s a n o r m a l e x i t MTS,  with  Acronyms,  no  restart  especially  unavoidably  pretentious.  economy  technical  in  supervisor  will  4.1.3  Case:  Test  Meyers* model has Forest  extensive  Service,  analysis  will  supervisor. supervisor  is  per  empirical  and  basal  a family stand  allows  and  simulation  clarity  optimization  SIMOFT.  whole s t a n d / d i s t a n c e use  section, on  a  United  problem  demonstrate the  to e v a l u a t e  optimization  the  cf  M e y e r s ' model w i t h  will  help  by  the  independent  optimization  the i n t e r a c t i v e  facilities  utility  analysis. A brief  States  of  of  the  Meyers'  o u t l i n e of  Meyers'  necessary.  state  an  this  and  are  r e f e r r e d to as  operational  exercise  system  algorithm  area  be  The  use  situations,  Model  performed  This  i n an  basal  In  be  approach  The  However, t h e i r  (1971) e m p i r i c a l ,  had  academic  writing.  Meyer's  supervisor  capability. in  periodically  from the  area.  v a r i a b l e s of acre  and  stand  relationship Results  for  stand  age,  of  and  the  between two  cf thinning  cf curves r e l a t i n g  diameter  M e y e r s ' model a r e  of these  studies are  basal area  standard  core  levels  the  stand  model  is  variables,  dbh  used t o  after thinning of  dbh,  growing  to  generate average  stock.  The  80  •growing unique  stock  level'  basal area  at  (GSL)  an  average  M e y e r s ' model s t a r t s prior and  to  age.  computed partial  computing  change  thinned the  stand  stand  dbh,  basal  dbh  and  basal  Meyers' ponderosa modified The the  per  optimization model was  Minimum  commercial cut  roundwood s a l e s and  log  operations.  function  values  as  100  index  and  age. fi  simulated  due  a  thinning,  the  thinning  a  function  fixed  as  by  to  volume b e f o r e  as  is  a  and of  prediction  function  of  calibrated  were  only  for  slightly  environment. with  f o r saw  cubic  economic  Meyers logs  model  measures o f t h e  an e c o n o m i c model (1973)  was  300  f e e t as  (Table  1500  cubic  returns  four a  based 1) . 8  FBM/acre.  f e e t per  a byproduct  performance of  The s t a n d models and e c o n o m i c d a t a selected to illustrate technigue, comment on t h e i r v a l i d i t y . 8  stand  for  o f roundwood was  from  The  height  Laws.)  combined  cut  is  programs,  b e n e f i t assumptions of  A minimum c o m m e r c i a l  stand  acre.  ponderosa  and  level  index  calculating  site  {Pinus  stand  cost  and  young  codominants  site  m o r t a l i t y i s computed  area  inches.  and  growth i s computed  computer  f o r the  and  and  original  pine  of  between t h e  area,  Moncatastrophic  dominants  diameter  difference  is its  average diameter, s i t e  growing stock  volume,  t h i n n i n g . Diameter  period.  of  in  10.0  a d e s c r i p t i o n of the  functions  to a s p e c i f i e d  as  previous  on  height  f o r each curve  diameter of  density,  regression  the  computing  after  average  from cut  volume  with  t h i n n i n g : stand The  designation  from  acre saw  objective management  sets in this thesis were and are presented without  81  sequence.  Table  -  maximize volume p r o d u c t i o n  (CCF)  -  maximize volume p r o d u c t i o n  (MBF)  -  maximize d i s c o u n t e d  -  maximize  discounted  rotations  ($)  1,  net  v a l u e , one  net  value,  C o s t and b e n e f i t a s s u m p t i o n s a n a l y s i s o f M e y e r s ' model.  rotation infinite  used i n t h e  ($) series  of  optimization  Costs Preccmmercial t h i n n i n g C l e a n u p (no s a l v a g e ) Saw Log S a l e Bound wood s a l e Seeding annual c o s t Prices Bound wood stumpage Saw l o g Saw l o g s from t h i n n i n g  Due variables  to  the  nature  that take  on  2.50 15.00 12.75  /CCF /MBF /MBF  continuous  The  ages  must  interval  c f p r o j e c t i o n , ten  years  o p t i m i z a t i o n problem  thinning  $/acre $/acre $/MBF $/CCF $/acre $/year  Meyers*  levels.  An  cutting  of  25.00 25.00 1.56 0.05 30.00 0. 20  involving  values be  forest  intensity  of  manager thinning  are  the  wishes  the  integer  f o r the  regime i s d e s c r i b e d i n Meyers'  A  model, the  1971  that w i l l  growing  stock  multiples cf  ponderosa  growing  to  only d e c i s i o n  pine  stock  determine maximize  model.  levels  paper:  the  volume  the  of  a  82  production 70,  i n board  Alternatives  precommercial commercial  thinning  the  regeneration  that  c o n t a i n s 950 t r e e s  average  feet.  of  low  produce The  the  response  surface  initial  and  yielded  28 MBF.  produced  29.3  This supervisor. free  volumes  of  with  one  maxima  levels  found  to see that  qrowinq  stock  w i t h one p r e c o m m e r c i a l  thinning  initial  found  subsequent  at  levels, thinning. (80,80),  respectively,  was  at  that  (120,100)  Although  i t i s somewhat u n i n t e r e s t i n g ,  solved  below  of  s e r v e t o demonstrate  computer  the  The  session  starts SIMOPT  the  and  mode  w i t h o n l y two  economic  user session cf  model,  as  presentation  the  uppercase performed will  be  with e x p l a n a t o r y t e x t .  with the user i n i t i a t i n g and a s s i g n i n g  optimization  t h e t e c h n i q u e . The  i s the a c t u a l  terminal.  using  user s e s s i o n s i n t e r s p e r s e d  t h e program always  was  is  p r o v i d e d below  is  with  MBF.  material  of  stand  30,  problem  demonstration  that  were t h e n examined  intermediate  The o t h e r maxima  will  The  i n a new  and low s u b s e q u e n t  v a r i a b l e s and no r e q u i r e m e n t  a  result  which  was b i m o d a l  problem  at  Minimum  t o a 4-inch  requirements. I t  and  subsequent  one  t a b l e s of combinations  initial  greatest  than  The manager e x p e c t s  levels,  l e v e l s o r of hiqh i n i t i a l  feet  index  o f 4.8 i n c h e s .  o f them met t h e p r o b l e m  combinations  more  p e r a c r e by age  50 y i e l d  thinning  site  unacceptable.  cuts w i l l  diameter  Meyers g e n e r a t e d  which  are  for  volumes a r e 320 c u b i c  and 1500 b o a r d  subseguent  i n stands of  calling  top  an  and  feet  some f i l e s .  a s s i g n e d and c o n t a i n s a s e l e c t i o n  the  execution  The POLICY  of i n i t i a l  file  policies  83  for  the various  -POLICY w i l l  guestion  will  store  the  managed s t a n d  model p r o d u c e s . The s u p e r v i s o r mark  POLICY PP#1  yield  prompts  file  policy, tables  for input  and which  with  a  (?) .  # $RUN SIMOPT 3=P0LICY # EXECUTION BEGINS ? READ PP#1 3  ?  The t e m p o r a r y  be u s e d t o s a v e an i n t e r i m o r optimum  -YIELDTABLES Meyers'  models i n t h e model l i b r a r y .  7=-P0LICY  , 19 PARAMETERS  8=-YIEIDTABLES  , 11 VARIABLES  ARE READ I N .  DISPLAY  PP#1  TEST CASE  : MEYERS MODEL  PARAMETER VALUE #  D E S C R I P T I O N  6 2.00 80.00 7 8 320.00 9 1500.00 10 10.00 11 70. 00 12 30.00 13 950.00 4. 80 14 15 25.00 16 25.00 17 1. 56 18 0. 05 30. 00 19 20 0. 20 2. 50 21 15.00 22 23 0.02 24 2.00  MEYER'S MODEL BASE GSL MIN KERCH CF VOL MIN MERCH VOL BF INTERVAL OF PROJECTION SITE I N I T I A L AGE TPA AT THIN STAND DEH PRE COM THIN COST /ACRE CLEAN UP COST -NO SALVAGE TIMEER SALE COST /MBF TIMBER SALE COST /CCF SEEDING ANNUAL COSTS STUMPAGE /CCF STUMPAGE /MEF DISCOUNT RATE IOBJ = MAX:MBF  VARIAELE #  VALUE  1  2.00  STATUS 0  STEP  LOWER  UPPER  SIZE  BOUND  BOUND  0.0  0.0  0.0  D E S C R I P T I O N CYCLE  84  2 3 4 5 6 7 8 9 10 11  The the  0 0 0 0 0 0 0 0  policy f i l e  identify  the  parameter  24  i s the  t h o u s a n d s of  the  the  of  0  dimensions of explicit  and  the  free  to  the  The  label  PP#1  from  policy  is  then  those  elements of  decision  commands.  i s to  the  variables,  each p a r a m e t e r and  i s used  For  to  example,  The  value  maximize volume p r o d u c t i o n  it  on is  i n the  analogous i n t e g e r  The  new  well  simplex,  The and  earlier.  level  as  •status*  values  fixed.  growing stock stock  an  algorithm.  original  growing vary  the  performance c r i t e r i o n .  constraints described  above, only  3.  be  v a r i a b l e s , as  take  if  the  to  provides  the  simplex  1 i f i t i s f r e e to or  GSI AT FIRST THIN MIN GSL AFTER THIN AGE FIRST CUT %GSL AT ROTATION NEW INTERVAL #2 CUT SHELTERWOOD %GSL AT 2ND CUT INTERVAL #3 CUT SHELT ERWOOD 1ST THIN INTERVAL  2 in  feet.  values  seguential  of  objective  board  likely  supervisor  choice  variable l i s t  initial  process,  in  unit  defines  would n o t  parameter  indicates that  and  list  number i s a s s i g n e d  140. 0 140. 0 0. 0 0. 0 G. 0 0. 0 0. 0 0.0 0. 0 0. 0  p o l i c y with  to l o g i c a l  parameter  model which  integer  50.0 50.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  reads the  assigned  The  simulation  The  10.00 10.00 0. 0 0.0 0.0 0.G 0.0 0.0 0.0 0.0  1  READ command  DISPLAYed.  An  1  80.00 80.00 11. 00 0.50 2. 00 13. 00 0.0 0.0 0.0 2.00  after  in *step the In  identifier  information of  a variable i s  the  optimization  size  1  defines  * bounds' the  the  are  process.  the the  policy  DISPLAYed  first  thinning,  l e v e l s a f t e r subsequent t h i n n i n g s ,  optimization  for  are  ?  SIMULATE  30. 50. 70. 90. 110. 130.  PEECCMMEECIAL THIN ROUND HOOD SALE SAB LOG SALE SAW LOG SALE SAW LOG SALE SAW LOG SALE OBJECTIVE  The values a  The  reports The  stand  the economic yield  foot  table  volume  sawtimber. timber. -16.10)  yield  roundwood cuts  round  harvesting  costs  0 1 2 3 4  Z* 27.94 2 29.395 30.000 30.000 30.040  basis.  and  f i l e -YIELDTAELES,  report  at the  (Policy  defray  (>320  i s  terminal.  #1). From t h e precommercial.  cubic  yield  merchantable  wood  sale  exceeded  the cost  was  feet)  cubic b u t no  volumes o f saw  unprofitable  ($  b e n e f i t s . The s a l e o f t h e  of the thinning cut.  V A R I A B L E S 2 3 80. 00 80.00 80. 00 90. 00 90. 00 70. 00 70.00 90.00 75.00 92. 50  dollar  M e y e r s ' model p r o d u c e s  a t age 30  sale  ? OPTIMIZE 30  #  -50.0 -16.1 24.9 38.8 124.6 172.4  a t age 50 p r o d u c e s enough m e r c h a n t a b l e  the  roundwood h e l p e d  A l l volumes  i n Appendix V  the thinning  The s u b s e q u e n t  as  $  362. 2 751.0 865.3 2114.1 2007.5  t a b l e on t h e s c r a t c h  i s included  for a  Note t h a t  above.  model p r o d u c e s a s h o r t  thinning  EETURN  27.94  on a p e r a c r e  one s e e s t h a t  second  0.0 1650.0 3466.8 106 37.6 12187.8  i s SIMULATEd  are expressed  managed  and  VALUE IS  policy  VOLUME OF CUT CF. BP.  AGE  A C T I V I T Y  5 6 7 8 9 10 11 12 13 14 15  75.00 73.75 74. 38 74.38 73. 91 74. 53 74.53 74.53 74. 53 74.53 74. 53  30.040 30.107 30.195 30.195 30.310 30.356 30.356 30.356 30.356 30.356 30.356  92. 50 91.88 92. 19 92. 19 92. 58 92. 89 92, 89 92.89 92. 89 92. 89 92. 89  AFTER 15 ITERATIONS,THE BEST RETURN IS FREE VARIABLES : 74.53 92.89 ?  SIMULATE  A C T I V I T Y  AGE  PRECOMMERCIAL THIN PREC CM MEBCIAL THIN SAW LOG SALE SAW LOG SALE SAW LOG SALE SAW LOG SALE  30. 50. 70. 90. 110. 130.  The  VOLUME OF CUT BF. CF.  1568.0 3547.9 11288.1 13952.0  OPTIMIZE p r o c e d u r e was i n v o k e d  iterations  but c o n v e r g e n c e  objective  function  of  had o c c u r r e d  maximizing  27.942 t o 30.356 MBF  as t h e p o l i c y  subseguent  80  levels  GSL  o f 74.5 and  demonstrates violating below  of  to  (variable years.  i t  t h e problem increase 11) from  and  92.9.  that  RETURN $ -50.0 -50.0 25.4 39.7 132.8 199.3  794.5 924.4 2313.8 2355.4  30. 36  OBJECTIVE VALUE I S  70  30.356  80  changed  maximum  of  30  15 i t e r a t i o n s .  The  two  interval  cf  increased  from  initial  and  from an  respectively,  specifications. the  after  a  volume y i e l d s  Simulation  involves  for  this  t o growing  stock  optimal  policy  precommercial The SET after  command  the  2 t o 4 decades, e l i m i n a t i n g  thinnings,  first the  is  used  thinning  thinning  at  87  ? SET V 11 4. ? OPTIMIZE 30 i A B L E 2 3 74. 53 92. 89 74.53 92. 89 95.39 79.53 75.7 8 98. 52 75.76 98.52 75.78 98. 52 75.78 98.52 75.78 98. 52 75.78 S8. 52 75.70 99. 41 75.70 99.41 76. 28 100.29 76.28 100.29 75. 94 99. 61 99.64 76. 10 76. 10 9 9. 64 99.64 76. 10 99. 64 76. 10 99.64 76.10 99, 54 76. 01  v a #  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19  Z* 30.011 30.011 30.075 30.950 30.950 30.950 30.950 30.950 3C.950 31.139 31. 139 31.226 31.226 31.237 31,237 31. 237 31.237 31.237 31.237 31. 237  R  AFTER 19 ITERATIONS,THE BEST RETURN IS FREE VARIAELES : 76.10 99.64 ? SIMULATE  A C T I V I T Y PRECCMMERCIAL THIN SAW LOG SALE SAW LOG SALE SAW LOG SALE SAW LOG SALE OBJECTIVE VALUE I S ? WRITE 7 ? STOP # EXECUTION  30. 70. 90. 110. 130.  VOLUME OF BF. 1621.0 3263.7 1 1724.1 14627.8  CUT CF. 923.9 917.0 2532.7 2539.2  RETURN $ -50.0 28.1 36.5 138.5 209.6  31.24  TERMINATED  Further age  AGE  31.237  70 r e s u l t s  optimization  o f the p o l i c y  i n an a d j u s t m e n t  o f the  without the t h i n n i n g a t initial  and  subseguent  88  growing  stock  levels  t o 76 and 99.5 s g . f t . ,  an improvement i n y i e l d shows  only  one  profitable resulted stock in  in  levels  Appendix The  t o 31.237 MBF. S i m u l a t i o n  precommercial t h i n n i n g  saw  log  a  sales.  best  V  the  STOP command. problem  was  function the  further  from  net  p r e s e n t worth.  (80,80),  s e t at 4 decades.  a subseguent  analysed  maximizing  -chinning  with  table  analysis  is  included  and i n i t i a l  the  file  with  i s terminated with  after  board  changing  the  yields  to  foot  The p o l i c y interval  The o p t i m i z a t i o n  was r e l o a d e d and after  the  algorithm  GSL o f (115,108),  first  converged  with a present net  o f $27.13 p e r a c r e . Knowledge o f t h e o p t i m a l  information behavior  for  respect  sensitivity  to the decision  command.  sq.ft.  the decision  of the o b j e c t i v e  A rough  SET  c u t s as  29.3 MBF a t g r o w i n g  and e x e c u t i o n o f s u p e r v i s o r  from  worth  produced  Meyers'  i s t h e n s a v e d on t h e t e m p o r a r y  optimized  at  policy  ( P o l i c y #2).  WHITE command  maximizing  comparison,  that  of the  with subsequent  o f 120 and 100 s g . f t . The y i e l d  the  objective  In  policy  optimal policy  The  r e s p e c t i v e l y , and  The  Simulation objective  runs  were  function There  is  usually  maker; one a l s o  function  at points  line.  First,  the optimization  nearby, function  v a r i a b l e s c a n be a c c o m p l i s h e d thinning  GSL was v a r i e d performed  GSL  are  two  was  at 1 sq.ft.  with the  a t 108  108 - 117.  i n t e r v a l s and t h e Figure  features of i n t e r e s t  algorithm  with  Fixed  o v e r t h e range  values are represented i n  solid  insufficient  needs t o know t h e  a n a l y s i s o f the o b j e c t i v e  subsequent  and t h e i n i t i a l  policy  has c o n v e r g e d t o a  8  with  a  i n Figure  8.  suboptimum  89  1  1  105  1  1  107  1  i 109  i  1  111  1  1  113  Intensity of f i r s t (Growing F i g u r e 8.  1  1  115  1  1  117  1  1  r  119  thin  stock l e v e l after t h i n n i n g )  S e n s i t i v i t y o f Meyers' model t o r a n g i n g o f t h e i n t e n s i t y of the f i r s t t h i n n i n g .  90  point  (115,108)-  while  (111,108).  Second,  apparently  has  the  two  at  least  surface  sharp  one  of  better  the  objective  discontinuities,  (116,108).  One o f t h e s e 'deep r a v i n e *  the  o p t i m a , f o u n d by t h e o p t i m i z a t i o n  local  better  point  at  convergence. from the  (111,108)  The  nature  the s i m u l a t i o n thinning  and  point  at  function  (112,108) and  discontinuities  may  separates  algorithm,  account  for  a t age 70 y i e l d s 1742.7 b d . f t .  optimum of  from the  the  o f t h e d i s c o n t i n u i t y can be  t a b l e s . At the a p p a r e n t  exists  false  discerned (111,108),  sawtimber:  ? SIKULATI  A C T I V I T Y  AGE  P B EC CM M EEC I AL THIN SAW LOG SALE SAW LOG SALE SAW LOG SALE SAW LOG SALE  30. 70. 90. 110. 130.  $34.00 the  the  per  first  thinning  i s sufficient acre  thin  1742.7 4481.4 11561.8 14473.7  i s produced.  sq.ft.,  i s r e d u c e d t o 1493 b d . f t . , f o r a sawlog  $3.40 p e r a c r e :  sale  and a r e t u r n o f  However, by i n c r e a s i n g  sale  a l e s s p r o f i t a b l e roundwood s a l e only  -50.0 34.1 50.1 136.8 206.7  1211.8 1019.1 2688.8 2639.4  volume f o r a s a w l o g  by 1 t o 112  minimum l i m i t  SETUBN $  28.43  OBJECTIVE VALUE IS  This  VOLUME OF CUT BF. CF.  the which  yield  of  t h e GSL a t the  is slightly  second  l e s s than  (1500 b d . f t . ) . C o n s e q u e n t l y ,  was s i m u l a t e d ,  which  returned  91  ?  SIMULATE  A C T I V I T Y  30. 70. 90. 110. 130.  PRECCMMERCIAL THIN ROUND WOOD SALE SAW LOG SALE SAW LOG SALE SAW LOG SALE  by the  SETting  discontinuity  ? OPTIMIZE  #  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  was removed from  t h e minimum b o a r d  optimization  1493.6 4262.6 11531.0 14393.3  -50.0 3.4 47.7 136.2 205.3  1157.9 1028.1 2690.6 2644.9  algorithm  foot  was  the o b j e c t i v e  volume c o n s t r a i n t  restarted.  30  Z* 27.579 27.579 27.579 27.57 9 27.579 27.579 27.579 28. 248 2 9.089 29.089 29.089 29.089 29.089 29.089 29.089 29.089 29.089  RETURN $  11.09  OBJECTIVE VALUE IS  This  VOLUME OF CUT CF. BF.  AGE  V A R I A 2 115.00 115.00 115.00 115.00 115.00 115. 00 115.00 110. 00 11 1.09 111.09 111.09 11 1. 09 11 1.09 111.21 111.21 111.10 110. 90  B L E S 3 108.00 108.00 108.00 108. 00 108.00 108. 00 108.00 108.00 108. 31 108.31 108.31 108. 31 108. 31 108.48 108.48 108.44 108. 35  AFTER 16 ITERATIONS,THE BEST RETURN IS FREE VARIAELES : 111.10 108.44 ? SIMULATE  29.089  function  to zero,  and  92  A C T I V I T Y PRECOMMERCIAL SAW LOG SALE SAW LOG SALE SAW LOG SALE SAW LOG SALE  VOLUME OF BP.  AGE  THIN  30. 70. 90. 110. 130.  OBJECTIVE VALUE IS  t h e a p p a r e n t optimum  analysis surface  was  repeated  might  conceive  of  optimized. intensity  For  problem the  but  The  converged sensitivity  r a n g e and  the  smooth  line.  number  and  is  of  ways  initial yield  decision  this  difficult  t h e n one yields  available  for  the d i s c c u n t  optimum  r e s u l t s i n high  is  variables  thinning  because  A second  not  seguence  second t h i n n i n g ,  the  thinning  might  being are the  can  imagine  low  guality  the  second  rate favours  occur  quality  to  and  where  a  high  value  Meyers  test  thinning.  multimodal resulted  the  desirable  returns.  are  management  i f  where a heavy  first  economic  a  functions  stand  example,  wood i n t h e s e c o n d The  #4).  8 with the s o l i d  demonstrates  reducing  thinning  moderate  a  c f the f i r s t  optima  thinning,  V, P o l i c y  the o r i g i n a l  objective  where  wood, g r e a t l y  early  over  algorithm  fail.  Multimodal  a local  (Appendix  above e x e r c i s e  approach  -50.0 33.4 51.4 137.9 208.9  1188.9 1036.6 2701.8 2675.7  removed, t h e  i s represented i n Figure  The  RETURN $  29.09  With t h e d i s c o n t i n u i t i e s at  1674.5 4591.2 11666.1 14621.6  CUT CF.  situation  described  from a d i s c o n t i n u i t y  model.  Models  that  are  in  the  c a u s e d by a c o n s t r a i n t i n to  be  analysed  by  this  93  approach  must  variables,  be  behaved stand  for  model  sq.ft. Of  in  results  thinning  carefully  or state t r a n s i t i o n  discontinuities acceptable  examined  the  optimization. that  of  a  year  Meyers  1  a  model  guestion  stand,  of as  intensity  a  Models t h a t  Case: G o u l d i n g ' s  Goulding's Douglas-fir Chapter  a  a  1  a t age  30.  in this  2  as  an  Model  growth model  was  characterized  empirical, single tree, distance  evaluate  to  optimization  i t s utility  in  s t a t e v a r i a b l e s o f t h e growth dbh  in  independent  processes  conjunction  is  with t h e  supervisor.  vector  a 1/4 a c r e  of  a  plot)  . The s i m u l a t i o n  by g e n e r a t i n g  the  e l e m e n t s o f t h e dbh s t a t e  distribution. height  an a v e r a g e s t a n d  Each  model a r e t h e  representative  20  dominant  at  t o d e m o n s t r a t e some o f t h e  necessary  (usually  of a  of  t o be used  d e s c r i p t i o n o f some o f t h e model's  a  enough  bd.ft.  result  model. A b r i e f  and  give  the v a l i d i t y 250  in  c f the approach.  4.1.4 T e s t  The  result  o f the stand  was n o t d e s i g n e d serves  will  logical  may n o t be w e l l  reduction  old  manner, b u t t h e above a n a l y s i s pitfalls  function,  One might  change i n t h e t h i n n i n g  course,  that  simulations  predicts 70  functions  objective  as s i m p l e  for constraints,  cycle  o f t h e stand  as  growth a  is initiated  according begins  function  to  age  of the stand at  dbh and t h e n a s s i g n i n g  vector  of  sample  stand  an  age  dbh t o  empirical  by c o m p u t i n g t h e of  age  and  site  94  index, The e x p e c t e d empirical  mortality  regression  including  age, s i t e  function  index,  Goulding  adds a s t o c h a s t i c  deviate  from  error of  the  mortality  is  dbh s t a t e  then  vector.  average  regression dominant  elements  of  calculations six  of  dbh  i s also the  stand  average  operations removes until  increment  random  This  repeated  mortality  i s also  basal  area,  i s then  total  among  computed  the  site  from a index,  and stems per  allocated  increment  by  process.  acre.  between  i n a c o n t i n u o u s manner,  the  according  and  f o r each growth p e r i o d  p r o v i d e s f o r two  on t h e dbh v e c t o r .  trees  in  a desired  The f i r s t ,  ascending  amount o f  modes o f t h i n n i n g ,  order  basal  s e c o n d , t e r m e d a crown t h i n n i n g ,  mortality  of l e s s  than  termed  both  involving  a low  thinning,  from a s p e c i f i e d minimum  area  has  been  removes t r e e s  removed.  from a  dbh The  predefined  class. The  major  optimization mortality this  a  acre.  by t h e s t a n d a r d  a stochastic  stand  an  years. Goulding  dbh  by s a m p l i n g  characteristics;  dbh,  vector  of  r e l a t i o n s h i p . The dbh  are  per  relationship.  allocation  for  the state  t o an e m p i r i c a l  stems  d i s t r i b u t i o n defined  vector  function  The a v e r a g e s t a n d  from  characteristics,  and  formed  regression  The  dbh  height,  stand  area  component  i s computed  d i s t r i b u t e d between t h e stems r e p r e s e n t e d  elements of the s t a t e The  of  basal  an e m p i r i c a l  the  mortality  (stems p e r a c r e )  modification  analysis  component.  stochastic  was  to  Goulding  component  made  to  remove  Goulding*s the  model  random  for  residual  determined  that  the i n t e r a c t i o n of  of m o r t a l i t y  with  the other  elements  95  of  the  model r e s u l t e d i n a s m a l l  allowed this to  that  one,  'the  and  obtain  of  additional  the  stochastic  model  framework.  result  stochastic needed  is  was  an  i n a wholly  of  into  subroutines  s t a n d s , grow s t a n d s t o and  The  stand  model b a s e d (Table  cubic  converted  to  original but  the  cf  the  optimization  m o r t a l i t y among t h e  stems o f t h e  dbh  of  the  could  not  be  random r e s i d u a l m o r t a l i t y  does  not  age,  model  that  model. were  points perform  utilization  generally  to c r e a t e  twenty year  thinning  or  clear  old  cuts,  standards.  g r o w t h model was  combined  Hoyer's  cost  (1975)  reorganized  and  with  a simple  benefit  economic  assumptions  2) .  The  factor  cn  used  the  entry  compute volumes t o any  an  as  variability,  programs  any  unclear,  in  component  deterministic  with  he  of  the  Goulding's o r i g i n a l  f o r the  evaluations  undesireable  i n t e g r a l part  elimination  is  repeated  second  a l l o c a t i o n of  However,  n e g l i g i b l e when  rationale  element  for  clearly the  would be  The  9  Because  probabilistic  removed,  values' .  effort  in yield.  between a s t o c h a s t i c model s u c h  a d e t e r m i n i s t i c one  average  inclusion  vector,  difference  decrease  to  of  thinning  » Goulding  foot  board 7. was  The  volumes produced  feet  by  assuming  an  minimum m e r c h a n t a b l e  constrained  (1972, p. 115)  to  be  by  Goulding's  average  lumber  volume t h a t  8 cunits  per  model were  acre.  was  reccvery feasible  96  Table  2,  C o s t and b e n e f i t a s s u m p t i o n s model (Hcyer (1975) ) .  Ease y e a r annual cost /acre stand estab. cost /acre precommercial t h i n cost c l e a r c u t s a l e c o s t /MBF t h i n n i n g s a l e c o s t /MBF p r i c e /MBF  The  e c o n o m i c model r e t u r n s  of Goulding's  1971 $ .92 $ 56.06 $ 54.5 8 $ 3.89 $ 8.75 14.05 + 1.34 dbh  /acre $  six  used w i t h  objective  function  values  as  m e a s u r e s o f p e r f o r m a n c e o f a management s e g u e n c e : mean  annual increment  of close  utilization  volume, i n  cunits mean a n n u a l i n c r e m e n t  of close  utilization  volume,  in  MBF mean  annual  increment  of  intermediate  utilization  of  intermediate  utilization  volume, i n c u n i t s mean  annual  increment  volume, i n MBF  The  present  n e t worth o f c l o s e  present  n e t worth o f i n t e r m e d i a t e  following  neighborhood management the  user  p o l i c y space sequence.  session of  a  The example  c h a r a c t e r i s t i c s o f Goulding's  optimization  analysis.  utilization  The  volume  utilization  explores  multiple  the  commercial  volume  optimal thinning  w i l l a l s o d e m o n s t r a t e some o f model  supervisor  in  the  framework  of  program i s i n i t i a t e d i n  97  exactly a  t h e same  scratch  tables  way  file  as with t h e Meyers' t e s t  i s assigned  generated  by G o u l d i n g ' s  # $RUN SIMOPT 3=F0LICY # EXECUTION BEGINS ? READ DF#1 3 POLICY DF#1  to l o g i c a l  unit  case,  except  9 to hold  that  the stand  model.  9=-STANETABIES  , 18 PARAMETERS  ,  7 VARIABLES  ARE READ IN.  ? DISPLAY  DF#1  TEST CASE #1  * «••• ••••••  : GOULDING'S MODEL  •  •  •••• • •* •  ••  PARAMETER # VALUE  D E S C R I P T I O N  6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23  MODEL TYPE SITE INDEX PLOT SIZE DBH CU DBH IU DISCCUNT RATE ESTABLISHMENT COST STAND STOCKING CONTROLANNUAL COST CLIARCUT COST /MBF THIN COST /MBF MIN MERCHANTABLE CCF/ACRE DBH < 0.75 0.75 <= DBH < 1.00 1.00 <= DBH < 1.2 5 1.25 <= DBH < 1.50 1.50 <= DBH OBJECTIVE : MAX MAI CU  1, 00 150. 00 0. 25 7. 10 11. 10 0.06 56.00 54.00 0.92 3. 79 8.75 8.00 0.10 0. 25 0.32 0. 25 0.07 1.00  VARIABLE # VALUE 1 2 3 4 5 6 7  100.00 40.00 7.00 90. 00 20.00  io.oo  90. 00  STATUS 1 1 0 0 1 0 1  STEP SIZE  LOSER BOUND  UPPER BOUND  D E S C R I P T I O N  25.00 3.00 2.00 0.0 5.00 2.00 5.00  50.0 21.0 1.0 0.0 15.0 3.0 30.0  400.0 100. 0 4. 0 0. 0 30.0 15. 0 120. 0  STOCKING AT AGE 20 AGE I N I T I A L THIN MAX # OF THINNINGS MAX AGE OF THIN INTENSITY THIN %EA CUTTING INTERVAL AGE HARVEST CUT  98  The p o l i c y Douglas-fir discount  BEAD i n and DISPLAYed  stand.  Present  of  each  thinning  distributed given  in  variable 15%  plot,  and  of close  five  utilization  are  stand  the  basal  density  at  t o the i n t e r v a l  thinning  (40 y e a r s ,  t h e age o f t h e h a r v e s t  SIMULATE screen  command and  -STANDTABIES. The s t a n d  the  site  is  volume. The of stand  t o the  area.  The  age 20  (100 t r e e s  c u t (90 y e a r s ,  intensity  i s a free  returns stand  table  free  p e r 1/4  acre  the  yield to  t a b l e i s i n Appendix  age  of  t o 21 - 100 y e a r s ) ,  i s fixed  the  between  three  ccnstrained  between c u t s  the  proportions  other  50 - 400 t r e e s ) , constrained  a  b a s a l area i s  18 - 22. The t h i n n i n g i n t e n s i t y  120 y e a r s ) . The t i m e i n t e r v a l  ?  performance  i n t h e o p t i m i z a t i o n a n a l y s i s but i s c o n s t r a i n e d  initial  terminal  of  dbh c l a s s e s a c c o r d i n g  parameters  constrained  The  index  ( v a r i a b l e 5) i n p e r c e n t  and 30% o f t h e  variables  the  among  a high  n e t worth i s t o be c a l c u l a t e d w i t h  r a t e o f 6%. The c u r r e n t  mean a n n u a l i n c r e m e n t  above d e f i n e s  a t 10  table  the VI  to  30 years.  to  scratch  (Policy  the file  #1).  SIMULATE MAIN CBOP SI 150  *  GBOSS  *  THINNINGS  * * * * i | n t M * * * * * * i * * # * * * * * * * * * * * * * + ***** ** ********** ************* 4.7 20 400 53. 901. * 901. 45 * 40 216 10.2 136. 4634. * 5895. 147 * 8.3 39. 1261. 92 50 54 97. * 8146. 1 63 * 57 10. 4 37. 1388. 143 12.8 141. 97 15.2 136. 41 12.4 5809. * 10051. 168 * 60 38. 1593. 6 4 18.0 124. 56S6. * 11569. 165 * 70 28 14. 6 36. 1631. 6 3 20.7 161. 90 8182. * 14055. 156 * OBJECTIVE  VALUE IS  1.21  VALUE IS  1.30  ? SIMULATE 5 OBJECTIVE  99  The  stand  and  yield  tables  simulation  and t h e o b j e c t i v e  CCF/year.  Sfhen t h e p o l i c y  different average at IU  random number  MAI i s f o u n d  value  The worth  feasible  was  the r e s u l t  calculated  i s SIMULATEd w i t h  strings,  no t a b l e s  t o thinning  iterations. repetitions  CO  volume. The  last  The 20  policy  was  iterations  that  then were  the  V A R I A B L E S 1 2 100.00 40,00 125.00 40, 00 118.75 42.25 121. 88 4 2. 63 149.22 37.66 149.22 37. 66 149.22 37,66 149.22 37,66 190.67 37.52 190. 67 37. 52 190. 67 37. 52 190. 67 37. 52 190.67 37. 52 164. 1 1 37. 54 164.11 37.54 164. 11 37. 54 169.80 36,99 169. 80 36. 99 169.72 36. 80 180. 19 37. 16 180.19 37.16  using the  t h e minimum  5 20, 00 20. 00 23. 75 24. 38 29. 84 2 9. 84 29. 84 29. 84 30. 00 30. 00 30. 00 30. 00 30. 00 28. 68 28.68 28. 68 29. 42 29. 42 29.79 29. 90 29. 90  net  present  OPTIMIZEd f o r 40 performed  per i t e r a t i o n .  Z* -37.836 -20.798 -20.017 -19.439 -9.085 -9.085 -9.085 -9.085 -8.140 -8.140 -8.140 -8.140 -8.140 -7.784 -7.784 -7.784 -7.245 -7.245 -7.145 -5.456 -5.456  1.21  the t h i n n i n g s  ? SET P 23 5. ? OPTIMIZE 20 # 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  be  (parameter 17).  i n d e x o f p e r f o r m a n c e was r e S E T t o be  of  to  single  a r e p r o d u c e d and  the constraint  i s 8 CCF  of a  5 repetitions  t o be 1,3 C C F / y e a r . Note t h a t  a g e s 40 and 50 y e a r s v i o l a t e volume  are  7 90. 00 90. 00 80.00 85. 00 78.75 78. 75 78.75 78. 75 65.70 65.70 65. 70 65. 70 65.70 70. 30 70.30 70. 30 71.42 71. 42 70. 81 68. 25 6 8. 25  AFTER 20 ITERATIONS,THE BEST RETURN IS -5.456 FREE VARIAELES : 180.19 37.16 29.90 68.25 ? OPTIMIZE 20 3  with  3  100  Z* -14.714 -13.189 -11. 392 -9.884 -9.884 -7.164 -7.164 -7.164 -7. 164 -6.649 -6.649 -6.003 -6.003 -6.003 -5.895 -5.895 - 5 . 895 -5.795 -5.78 8 -5.765 -5.765  #  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  V A B I A B L E S 1 2 5 180.19 37. 16 29. 90 180. 19 40. 16 29. 90 39.41 130. 19 29. 97 155. 19 39. 79 29. 99 15 5.19 39. 79 29. 99 38. 15 170.82 29. 97 29. 97 170. 82 38. 15 38. 15 170.82 29. 97 38. 15 170.82 29. 97 37. 13 153.75 29. 99 37. 13 153.75 29. 99 156.07 37. 59 29. 99 156.07 37. 59 29. 99 156.07 37. 59 29. 99 37.98 29, 99 155, 25 37. 98 155.25 29. 99 37. 98 155.25 2 9. 99 155. 66 37. 78 29. 99 37. 76 155.59 29.99 38.01 29. 99 155.14 155.14 38.01 29. 99  7 68.25 68. 25 72. 00 65. 13 65.13 68. 33 6 8. 33 6 8.33 68. 33 68. 48 68.48 68. 04 68. 04 68. 04 68.46 68. 46 68.46 68. 25 68. 26 68. 17 68. 17  AFTEE 20 ITEBATIONS,THF BEST RETURN IS -5.765 FREE VARIAEIES : 155.14 38.01 29.99 68.17 ? SIMULATE MAIN CBOP SI 150 AFTEE THINNING AV. TOTAL AGE STEMS DEH EA VOL  * GBOSS * THINNINGS * PRODUCTION * * TOTAL * AV. TOTAL * VOL MAI * STEMS DBH BA VOL  ************************************************************* 20 38 48 58 68  620 4.3 272 9.1 144 12.0 80 15.1 80 16.7  68. 136. 125. 109. 134.  1109. 4437. 4741. 4605. 6070.  OBJECTIVE VALUE IS  The  index  of  worth o f CU  volume.  iterations.  The  repetitions The  * 1109. 55 * 6266. 165 * 8698. 181 * 10569. 1 82 * 12033. 177  * * * * *  192 108 59  7.2 9.5 11.7  60. 58. 49.  1830. 2127. 2006.  -5.96  p e r f o r m a n c e was r e S E T t o be t h e n e t p r e s e n t The  last  20  policy  was  then  OPTIMIZEd  for  40  iterations  were  performed  with  3  per i t e r a t i o n .  initial  policy  had a v e r y  low  net  present  worth  of  101  $ -37.836, 60  years.  the  probably  due t o t h e u n p r o f i t a b l e  (If a t h i n n i n g  economic  calculate  model  r e s u l t s i n a volume below t h e  charges  the return  thinnings  the  on s a l e s . )  thinning  cost  were made i n t h e f i r s t  4 iterations.  stand  density  100 t o 149 t r e e s  increasing (30%  of  thinnings  the i n t e n s i t y of the thinnings stand  basal  area),  was i m p r o v e d .  t o have s t a b l i z e d w i t h reduction thinning the  age  reveals limit.  intensity of  stand  table  that  the  with  net  first  where  the  (profitable) to  thinning  reasonable would  the  first  constant  acre  was  isstill  limit  of the e a r l y  and  a  slight  The maximum  prescribed, t o 68  and years,  VI,  appears  that  eliminating  an i n t e r m e d i a t e  was s t i l l  a t 48 y e a r s  cutting interval  to  have  worth.  This  o f human i n t e r v e n t i o n i n  has c o n v e r g e d  above 38 y e a r s ,  #2)  e l i m i n a t i n g the  the net present  necessity  Policy  below t h e 10 volume  algorithm  to suspect  improve  the  thinning  thinning  upper  the p o l i c y appeared  (Appendix  worth. E x p l o r a t i o n  thin slightly  the  substantially  no way o f s y s t e m a t i c a l l y  present  the  p e r 1/4 a c r e , and  t o 38 y e a r s .  area)  policy  38 y e a r s w i t h o u t e n c o u n t e r i n g  lower the  basal  optimization  demonstrates  1/4  thinning  t y p e o f a n a l y s i s . The a l g o r i t h m  maximum, at  i t is  to  increasing  a t age 70.  first  the  b u t does not  profitability  per  reduced  forthis  a t 38 y e a r s  situation  cut  the thinning  stabilized,  this  (30% s t a n d  By  40 i t e r a t i o n s ,  155 t r e e s  harvest  Although  thinning  the  i n t h e age o f f i r s t  eliminating The  After  minimum,  t h e most d r a m a t i c c h a n g e s t o t h e  policy  a t age 20 from  a t 40 and  to  a  local  the thinning  policy  with  of the p o l i c y r e g i o n would  involve  unprofitable  would have t o be o f 10 y e a r s .  a  with  policies  and t h e n e x t delayed  The j o i n t  due  effect  102  of t h i s even  delay  and t h e d i s c o u n t  further.  surface policy  Consequently,  there  is  from t h e c p t i m i a l two t h i n n i n g that  is  knowledge o f t h e optimization these  r a t e would be t o r e d u c e  local  continuous stand  algorithm  and  model,  no  pathway  policy  to a  economic  i s necessary  NPw  on t h e  NPH  one-thinning  nondecreasing,  the  the  fin  model,  to recognize  intimate and  the  and i n t e r p r e t  optima.  ? SET V 2 48. ? SIMULATE MAIN CHOP SI 150 * GROSS * THINNINGS AETEB THINNING * PRODUCTION * AV. TOTAL * TOTAL * A?. TOTAL AGE STEMS DBH EA VOL * VOL MAI * STEMS DBH BA VOL **************************************************** 20 620 4.3 68. 1109. * 1109. 55 * 48 208 11.3 159. 5965. * 8533. 178 * 152 8.8 71. 2568. 58 117 13.9 137. 5704. * 10762. 186 * 83 11.2 62. 2490. 68 116 15.4 166. 7473. * 12531. 184 * OBJECTIVE VALUE IS  The SET command thinning  at  48  the net p r e s e n t  ? OPTIMIZE # 0 1  was  years.  volume o f t h e f i r s t that  7.09  used t o a d j u s t Simulation  the  age  of the p o l i c y  t h i n n i n g i s above worth i s g r e a t l y  the  increased.  V A R I A B L E S 1 2 155. 14 48, 00 155. 14 48.00  5 29. 99 29. 99  7 68. 17 68. 17  the  first  shows t h a t t h e  minimum  20 3  Z* 5. 496 5.496  of  limit,  and  2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  496 5. 653 7. 114 7. 114 7. 153 7. 230 7. 230 7. 230 7. 230 7. 230 7. 452 •7.452 7. 452 7. 452 7. 452 7. 452 7. 749 7. 973 9. 578  AFTER  20 ITERATIONS,THE  155. 14 155.14 148.89 148.89 152. 01 152.60 152.60 152. 60 152. 60 152.60 152. 57 152.57 152.57 152.57 152.57 152.57 152.70 152.56 152. 70  FREE VAEIAELES OPTIMIZE #  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  :  48. 00 48.00 46. 88 46. 88 47. 44 47. 17 47. 17 47. 17 47. 17 47. 17 47. 23 47.23 47, 23 47. 23 47. 23 47.23 47. 22 47. 19 47. 35  29. 99 29. 99 29. 99 29. 99 29. 99 29. 99 29. 99 29. 99 29. 99 29. 99 29. 99 29. 99 29. 99 29. 99 29. 99 29.99 29. 99 29. 99 29. 99  BEST RETURN IS  152. 70  47. 35  68. 17 7 0.67 70. 05 7 0.05 69. 11 69.64 69. 64 69.64 69. 64 69.64 68. 94 68. 94 68. 94 68. 94 6 8. 94 68.94 68. 58 68.56 67. 48 9.578  29. 99  67.48  20 3  Z* 9. 578 9. 578 9. 578 9. 578 9. 578 9. 578 9. 578 9. 578 9. 578 9. 578 9. 578 9. 578 9. 804 9. 804 9. 804 9. 804 9. 804 9. 804 9. 804 9. 850 9. 900  V A R1 I A B L 2E S 152.70 47.35 152.70 47. 35 47. 35 152.70 152. 70 47. 35 47. 35 152.70 152.70 47. 35 47. 35 152,70 152.70 47. 35 152.70 47.35 47. 35 152. 70 47. 35 152. 70 47. 35 152. 70 47. 27 152.11 47. 27 152. 11 47. 27 15 2. 11 152. 11 47. 27 47. 27 152.11 47. 27 152. 11 47. 27 152. 11 152.31 47. 25 152.18 47. 22  5 29. 99 29. 99 29. 99 29. 99 29.99 29. 99 29. 99 2-9. 99 29. 99 29. 99 29.99 29. 99 29. 99 29. 99 29. 99 29. 99 29. 99 29. 99 29. 99 29. 99 29. 99  7 67.48 67. 48 67. 48 67. 48 67.48 67. 48 67.48 67. 48 67. 48 67. 48 67.48 67. 48 67.34 67. 34 67.34 67. 34 67. 34 67. 34 67.34 67.31 6 7.30  AFTER 20 ITERATIONS,THE BEST RETURN IS 9.900 FREE VARIAELES : 152.18 47.22 29.99 67.30 SIEULATE MAIN CROP SI 150 AFTER THINNING  * GROSS * * PRODUCTION *  THINNINGS  104  AV. TOTAL * TOTAL * AV. AGE STEMS B EH EA VOL * VOL MAI * STEMS DBH ********************************************* 20 608 4. 3 67. 1101. * 1101. 55 * 47 216 11.1 159. 156 8.7 5914. * 8472. 179 * 57 120 13.4 131. 5424. * 10690. 187 * 8 8 11.3 67 119 14.9 161. 7164. * 12431. 165 * OBJECTIVE  VALUE IS  ? STOP # EXECUTION  More only net  i s included  4.1.5  Test  stand is  (40) o f t h e OPTIMIZE  model  more  economic  by K i l k k i  Also  model makes i t i d e a l  volume  Scots pine that  growth  (Pinus  simulated  examined  and t h e  stand  #4).  the  under  distance  and V a i s o n e n volume  published  with  the o p t i m i z a t i o n  independent,  whole  ( 1 9 7 0 ) . The s t a n d  increment f u n c t i o n a complete  and  model but i s  detailed  s i m p l e and c o n t i n u o u s n a t u r e o f t h e analysis.  are the c u r r e n t fellows  sjlvestris  t a k e s the form  was  for optimization  The s t a t e v a r i a b l e s stand.  be  than a s i n g l e  model.  results i n  Model  to  here as i t was  algorithm  and some improvement i n t h e  (policy  program i s an e m p i r i c a l ,  m o d e l , used  included  policy  i n A p p e n d i x VI  Case: K i l k k i ' s  last  little  i n the p o l i c y  w o r t h . The f i n a l  table  supervisor  2558. 2709.  TERMINATED  small refinements  The  72. 67.  10.09  iterations  present  TOTAL BA VOL  1.)  volume and age  a growth f u n c t i o n by K u u s e l a and  of  the  calculated for Kilkki  (1963),  105  ,b, cv  -r  n  I = at where  10  v  I = c u r r e n t volume g r o w t h , cu.m./ha/year t  = age,  v  =  volume, cu.m,/ha  a,b,c The  are  constants  remainder  relationships curves  years  and  o f age  the  and  the  of growing  value  concerned  stock  assumed  cut  is  to  above, the of t h e  volume  and  cutting  Kilkki with  and  c o s t s are  growing  increment  performance  Vaisanen's  stock  be is  managed a c c o r d i n g Several  from  below,  value  value of  of the  regime  model and  the  a  the  the  growing value  growing s t o c k removed.  volume  and  stock  of  the  does  not  Thinning  value  of the  costs  removals  model was  level The  programmed stand  and  independently.  returned,  the  present  20  by  com p a t a b l e from  an  means o f  the  of  Only net  be  grown  intensity  scheduled  to the  is  (cu.m./ha) a t age time  to  up  one  worth  to index  of an  was  five of acre  s u p p l i e d s e q u e n c e of a c t i o n s .  m u l t i p l e t h i n n i n g r e g i m e s were e x a m i n e d and  thinning  as  cut.  function.  t h i n n i n g s can  stock  of  thinning i s calculated;  t a b u l a t e d by  t h e o p t i m i z a t i o n s u p e r v i s o r . The  initial  economic family  growing  d e n s i t y . The  a l t e r a t i o n s i n the  be  the  after  change with  volume t o  the  represents the  be i n d e p e n d e n t same as  and  of the  clear  of  management. A  thinning  value  and  the  consists  of  with  before  I n t h i n n i n g from  is  of stand  unit  d i f f e r e n c e between t h e s e  removed.  model  volume.  In c a l c u l a t i o n s value  the  assumptions  expresses  function  of  chosen  to  method o f a n a l y s i s .  demonstrate the The  initial  nature  pclicy  had  a  two  of  the five  106  free the  v a r i a b l e s ; t h e age and i n t e n s i t y c f t h e two t h i n n i n g s , and age a t c l e a r c u t . The i n i t i a l  was  assumed  calculation algorithms  ?  to  be  6  cu.m./ha  of the present converged  growing and  n e t worth  stock  the  was  level  a t age  discount  251.  on t h e p o l i c y SIMULATEd  The  rate  20 for  optimization  below:  SIMULATE VOLUME BEFORE CUT  AGE 55 70 84  VOLUME REMOVED  289.00 289.00 195.00  PRESENT NET  The  122.60 167.40 195.00  WORTH :  simple  NET RETURN 776.70 1242.90 1691.08  1096.84  construction  of  t h e model and r e s u l t a n t  execution  t i m e c a n be e x p l o i t e d t o examine t h e  climbing  algorithm  command o f previously varying years, were  first  subroutine  age a t f i r s t  held  at  their  of present  The and  two  second  optimum  The  was  used  performed  previous  value  of  to  58  the other  link  years  to  a  simulations,  variables  The r e s u l t i n g  and smoothed  p o l i c y was t h e n a l t e r e d  the  between 50 and 60  values.  was c o n t o u r e d  SET  to  A l l ether  optimal  cf  The SUBROUTINE  10,000  thinning  respectively.  thinning  values.  that  and s e c o n d  net worth  thinning  neighbourhood.  program  and 70 and 80 y e a r s ,  respectively. their  supervisor  written  the  surface 9).  the  i n the optimal  pathway  fast  with and  (Figure  t h e ages a t 78  years,  v a r i a b l e s were F i x e d a t  107  A g e at second thinning Figure 9.  The pathway o f the o p t i m i z a t i o n a l g o r i t h m on a s u r f a c e o f p r e s e n t n e t worth. ( K i l k k i ' s model).  108  The traced  pathway on  the  (iteration  0)  algorithm  (58,78) three  are  with  follows  the  The  a small  the  takes  6  more  (Figure  size  (.5, .5)  sloping  of  up  steepest  the  shrinking  step  by  size  exploration until  optimum  is  policy  the  simplex  rapid acceleration  about  the  initial  reaches  the  near the 4,  optimal  convergence i n s i d e the  (55,70),  neighbourhood  at i t e r a t i o n s the  in  algorithm  (55,76) t o w a r d s to  point  (55,77)  f o u r t h . The  simplex  of  i s just  initial  seguential  and  from  the  the  First,  ascent  c o n t r a c t i o n of  The  9.  from  algorithms  12).  of  (57. 5,72. 5)  ridge  of  9)  (iteration  Figure  iterations  12,  optimization  starts exploring  and  stage  iteration  on  step  line  from t h e  final  optimum  algorithm  optimum. The  evident  map  evident  f i n d s the  the  characteristics  iterations  guickly and  of  as t h e  by  contour  to the  A number  occurs  followed  of  optimum  is  5  6.  and  neighbourhood  is  reached  at  mapped r e g i o n  at  (55.2,70, 1 ) . Figure surface of  the  as  the  complexity  o f even a simple  and  reasonably  irregularities  of  the  contouring discrete  9 a l s o demonstrates the  process, and  but  most o f  discontinuous  tabular  harvesting  well  surface the  elements i n the  costs  and  behaved  may  detail  of the  be  can  objective model. Some  artifacts be  of  the  attributed  to  economic model, such  minimum m e r c h a n t a b l e  volume  constraints. A  last  technique  demonstrated  with  programming.  In  variable values.  the this  i s t r e a t e d as At s m a l l  of  optimization  optimization type  of  supervisor  sensitivity  a p a r a m e t e r and  increments of  the  analysis is  to  be  parameteric  analysis, a decision  varied over  parameter, the  a  range  remaining  of free  109  decision optimum  variables policy,  analysis the  decision  with  optimization variable  programming  may  by  centre!  of of  the  parametric  type of s e n s i t i v i t y  i n t e n s i t y of t h e f i r s t  variables  t a k e c n optimum  a r e numbered  intensity  parameter  and  incremented  remaining  free  variables  might As  the only  first  constant.  o f t h e optimum  accomplished  SET  and  analysis  with  the  algorithm,  OPTIMIZE  and  command.  was p e r f o r m e d  decision  i s f i x e d , and t h e r e m a i n i n g  values.  The d e c i s i o n  A  on t h e age  In both cases the  of the f i r s t  variables and  free  o f age age  at  from in  100  the  thinning  i s treated  as a  to  cu.m./ha,  the  f o r example,  175  optimum manner  d i s c o n t i n u i t i e s i n the optimal  policy (Table  policy  decreases.  thinning  3 ) . There a r e  of a  increases,  increases, The age  fluctuate i n  space, such  from t h e e l i m i n a t i o n  and s e c o n d t h i n n i n g  second t h i n n i n g  one  variable.  optimization  thinning.  the i n t e n s i t y of the i n i t i a l  the  held  in  t h e FIX command t o remove a  i n a smooth and c o n t i n u o u s  result,  changes  1-4 and 11, r e s p e c t i v e l y .  the  radical  demonstrates  and s u b s e q u e n t t h i n n i n g s ,  When  response,  the  as a p a r a m e t e r  i n t e n s i t y of i n i t i a l  clearcut,  no  to  variables  be  using  application  treated  from t h e r a n g i n g  demonstrates the s e n s i t i v i t y  supervisor  from  t o m a i n t a i n an  ranging  function  a l l other  repeated  and  differs  v a l u e t o c h a n g e s i n one d e c i s i o n  Parameteric  variable  re-adjust  4.1.3 i n t h a t  of the objective  Parameteric analysis  and  to  programming  i n section  variable,  objective  allowed  Parameteric  described  sensitivity  are  at  while  as  thinning.  the  age  at  the i n t e n s i t y o f  clearcut  fluctuates  slightly. Similarly,  when  t h e age a t f i r s t  thinning  i s treated  as a  110  Table  3,  Sensitivity of the optimal two t h i n n i n g seguence t o v a r i a t i o n s i n t h e i n t e n s i t y o f t h i n n i n g . ( K i l k k i ' s model)  V 2 (Fixed)  MAX: M I  100.0 105. 0 110.0 115.0 117.5 120.0 122.62 125. 0 130.0 135. 0 145.0 155.0 165. 0 175.0  1094.9 1095. 0 1095.3 1096.7 1096.7 1096.7 1096.8 1096.6 1096. 0 109 5. 2 1094.8 1092.1 1090.3 1089.9  53.8 54. 1 54.6 54.9 54. 9 54.9 54.9 54. 9 54.9 55. 9 55.9 57.0 57.3 57. 1  P r e s e n t n e t worth (PNW) i n d i s c o u n t r a t e o f 2%, The g l o b a l optimum.  1  2  p a r a m e t e r and v a r i e d free  of the f i r s t  thinning  ages  the second t h i n n i n g  i n t e n s i t y o f the second  increasing Of  age a t f i r s t  course,  a  increases  in  (Table  shews a downward  4 ) . The  while  change o n l y  the  the  sliqhtly. trend  with  thinning.  parametric  a n a l y s i s may  as w e l l a s d e c i s i o n  the  two  thinning rate  be p e r f o r m e d  v a r i a b l e s . Table  policies rises,  as  increases.  As t h e d i s c o u n t  is  and t h e i n t e n s i t y o f t h e f i r s t  reduced  83. 3 84.5 84. 1 84. 1 84. 1 83. 1 84. 1 84. 1 84. 1 84.1 84.1 84.4 85.4 85.4  the trends  markedly,  and c l e a r c u t  model p a r a m e t e r optimal  187.6 184. 6 181.3 174. 5 170. 7 170. 6 167. 8 166. 9 166.9 166. 9 166.9 166. 4 160. 0 160. 8  changes i n p o l i c y  thinning  11  c a l c u l a t e d with a  between 50 and 60 y e a r s ,  intensity  The  VARIABLES 3 4  69.7 69.6 69.3 70. 2 70. 2 70. 2 70.2 70. 2 70.2 70. 2 72. 1 72. 7 76.8 77. 1  dollars is  v a r i a b l e s show no r a d i c a l  at  FREE  1  1  management the first  the  5  thinning  records  disccunt  t h e age a t f i r s t  with a  rate  thinning  i s increased.  111  Table  4.  Sensitivity of the optimal 2 thinning management s e g u e n c e t o v a r i a t i o n i n t h e age o f t h e f i r s t t h i n n i n g ( K i l k k i 's m o d e l ) .  V 1 (Fixed)  KAX:  111  50 51 52 53 54 54.92 55 56 57 58 59 60 1  2  At  FREE VARIAELES 4 3  2  4  10 82.5 1084.6 10 93.3 10 94.0 1097.4 1096.8 1096.9 10 93.3 10 93.0 1091.7 1088.0 1084.1  101.0 112.0 116.7 117.5 118.2 122. 7 123.0 129.2 132.0 132. 1 135.6 136.5  69.5 70.5 71. 0 70. 6 70. 0 70. 2 70.2 64. 9 7C.3 70. 3 70. 3 71. 4  185.8 177. 9 173. 8 170. 8 172.4 167. 2 167. 1 165. 6 165. 8 170. 8 169.3 168. 6  85.0 85.8 84.8 85.7 85.2 84.1 84.0 84.0 84.2 84.2 84.0 84.4  P r e s e n t n e t worth (PNI) i n d o l l a r s i s c a l c u l a t e d d i s c o u n t r a t e o f 2%. The g l o b a l optimum.  interest rates  4.1,6  Summary And  The  analysis,  models should  executing The  thinning  with a  i s eliminated.  Discussion  foregoing  simulation  fast  above 4$, t h e f i r s t  1 1  e x a m p l e s have d e m o n s t r a t e d that  are  to  be  that  subjected  forest  to  stand  optimization  have t h e f o l l o w i n g c h a r a c t e r i s t i c s :  -  simulation  model s h o u l d  be  as  simple  as  possible,  112  Table  5.  Optimum management seguences with thinnings, at increasing discount model)  MAX: PNJi  M i l  .01 .02 . 03 . 04 . 04 + 1  FREE VARIABLES 3 4  3246.4 62 1096.9 55 498.9 53 257. e 48 commercial t h i n n i n g s  two  the  resolution, and of  Mead the  search  time  that  function  validity is  for  i s proportional  with a  and  desired  minimized.  t h e number  needed  92 84 83 84  Nelder  of  evaluations  the  sequential  t o the sguare of the  variables.  -  Stochastic  much  execution  converge  i s calculated  of  (1965) r e p o r t e d  of free  deterministic  policy  so t h a t  objective to  number  constraints  U  74 102 160 122 70 167 138 69 165 144 69 169 are u n p r o f i t a b l e  P r e s e n t n e t worth (PKH) i n d o l l a r s d i s c o u n t r a t e c f 2% .  within  two commercial rates, (Kilkki's  e l e m e n t s add t o  points  on  information  elements should  the  effort  the o b j e c t i v e to  a  be a v o i d e d  needed  surface,  decision wherever  to  without  analysis.  rank adding  Stochastic  possible.  unimodal Multimodal but  care  objective should  be  surfaces'probably taken  not  to  can't  be a v o i d e d ,  create  multimodal  113  situations  with  penalty  constraints  or  aggregated  variables.  continuous  -  Discontinuities discrete  elements  constraints, These  objective  i n the  or  algorithm,  should  aggregate For  avoid  other  example,  the  and  cause f a l s e  creating  efficiency  of  intensity  of  a later  early  environment  system  s i z e of  the  decision  be  thinning  and  increase  show t h e  v a l u e and  space  of  controlled imagine  decrease the  for this  kind  i n batch  of  mode, w i t h o u t  necessity  analysis.  intelligent  direct  is  that  variables.  decision  advantageous to  intervention  the  the  variables  I t i s easy to  A major r a m i f i c a t i o n  as  or  convergence.  variable  h a v e been u n s u c c e s s f u l .  such  data  a  the  intensity  cut.  examples a l s o  optimizing  by  of  counteracting  a l l thinnings.  where i t would an  decision  and  reduce the  situation  the  caused  relationships.  the  independent  to  intensity  interactive use  functional  may  be  tabular  reduce  G o u l d i n g ' s model, a s i n g l e  The  can  -  One  of  surface  model, such as  discontinuous  discontinuities  optimization  responsive  i n the  that  climbing  a submodel u n d e r c o n t r o l decomposition  approach  of  is an  suggested  of not  of  a  highly  Attempts  to  intervention,  this  need  for  suitable  for  automatic  process,  in Section  3.4.4.  114  4.2  ME2  As  A Network  Section decision a  3.4.1  problem  treatment  and  initial  can  a  value  to  continuous  be  discreet.  example. C o n s i d e r p r e d i c t e d by  is  will  and  be  curves years  nine  be  1.,  natural  section, is  be  e q u i v a l e n t to  the  A  transition.  of  u  u,  of the  the  all  as  a network,  treatment  stand,  where q r o w t h  to  two  intensities  density levels  and  state,  a t age  their  Discrete be  e x a m p l e , we  regeneration w i l l  described will  result  the  yield  20  of  stand  harvest  (cu.m./ha.)  years  (6., 4.,  are 3.,  model.  The  (Dl) a t age  20  corresponding  to  in  the  terms  stand  can  stocking  final  assume t h a t t h e in  simple  alternatives  the  density level states  must  o f t h i n n i n g . The and  the  It i s desired  .5 cu.m./ha.) u s i n g K i l k k i ' s  now  unit  different  C u r v e s r e p r e s e n t i n g volume  by  management  i s best d e s c r i b e d with a  in a wild  identified  solution  b e n e f i t s to each a r c , or  unit  planting  .75,  MP2  problem will  model, d e s c r i b e d e a r l i e r .  or  10).  value  programming  set  variables  a Scots pine  overmature,  (Figure  In  and  the  T h i s process  management a c t i o n s c a n curves.  initial  manaqement s e q u e n c e s i n c l u d e t h e  1.5,  will  for  dynamic  costs  three different  from  2.,  management s e q u e n c e  that  treatment  state  a clearcut.  generated 2.5,  cn  regeneration  currently  The  Kilkki's  the s e t o f  levels,  the  as an  problem  formulate  naturally  natural  t h a t MP2,  a p p r o p r i a t e model t o s i m u l a t e t h e  feasible  that  statement  programming. In t h i s  assigns  sequences  be  formulated  problem.  order  made  be  with the  the o p t i m a l  network  either  r e f e r e n c e s an In  of f i n d i n g  dynamic  as  algorithm  concluded  unit,  s o l v e d by  formulated  Problem  of  'activity' being  in  the DI of the  115  IT).  / j  J / DL =  6.0 / t '  t /  /  <  ,  /  /  '  '  J  '  m  5 o  Q  .  ,  in.  i  .  a  — CD. Qj CM  i  E  O 9  a a. 8  /  a.  r  '  '  /  /  /  t  ,  ,  J  •  /  '  /  j ,  j  '  ' '  J  >  /  '  t  '  -  />' •  '  ,  j' "  y  /  '  ,  ' ;  t  1-0/  '  '  t  /  f  ••/  ' /  - / 75  '  /  -t  '  0,5  '  / '  •  J  .  '  /  ,  / •>  y  ,  '  /  '  i  / / /  y  1  > ' i i i t  i i > t i ' ' '  '////  5 ' / 4 ' ->  i J  / '•  J  ,  '  /  ' ' • l  /  ' / //•/ <  7  in  i ,  i  i curve  /  i  / /  t  ,  '  y  < 2S  cn  /  ,  J  3.0  '  t ,  4.0' J  t  t  f  / /  o in.  j  ,  f  <  '  f  j t  /  <'  ,  ^  "  2 - *1 " 1 3 0 . 0  4 0 . 0  I 5 0 . 0  Age  F i g u r e 10.  BO.D (years)  7 0 . 0  8 0 . 0  S O . O  Volume/age curves c o r r e s p o n d i n g t o 9 levels of density a t age 20 y e a r s , o f Scots p i n e ( K i l k k i ' s model).  1 D D . 0  116  •state' while  of  having  planting will  a  result  naturally  regenerated  DL  cu.m./ha.  =  2  current The  density  from  These  states are l i s t e d  cyclic  graph t h a t  feasible  management  xn  actions  decade,  arcs  transitions  is the  the  precedence  (Figure  sequences  in  that  to  in  the  the current  the  All  arcs  time  relationships  represents  terms  and  wild  of  are f e a s i b l e  stand are  times  stand  graph  choice  of  of  the  may  be c u t any t i m e  will  management  i n Figure  represent stage.  length, five  state  A l l stages i n  years.  t h a t t h e management  be r e p e a t e d  assumed  graph  i s given  i s not r e p r e s e n t e d  The  cyclic  directly  but  seguence f o r  infinitely.  t o be d i r e c t e d , l e f t a management  management s e q u e n c e c a n be t h c u q h t the source  s e t of  f o r e a c h s t a t e and a t i m e  at a s p e c i f i c  are c f egual  o f an a r c r e p r e s e n t s  from  a  b u t does n o t i n c l u d e  timing  staged  t o the assumption  regenerated  feasible  cu.m./ha.  p r o v i d i n g i t i s c l e a r c u t by t h e end o f t h e  of the precedence  simplified  graph  11)  intensities,  and l a s t e n t r y  example p r o b l e m  choice  harvest.  by a d i r e c t e d  11, a c c o m p a n i e d  r e p r e s e n t a t i o n o f t h e precedence graph  The  nature  before  i n Figure  pertaining  d e c a d e . The f i r s t  the  change  is i t s  0.0  and t h e i r  F o r example,  the f i r s t  12.  to  wild stand  of  a c l e a r c u t i s , of course,  graph  management  information  staged  to the s t a t e  s t a t e of the current  represents  precedence  alternative  actions.  only  changes of s t a t e .  The  any  c a n be t h i n n e d  volume, and i s n o t e x p e c t e d  state resulting  years,  i n a s t a t e o f DL = 4 o r 2 cu.m./ha. A  stand The  o f 3 cu.m./ha. a t age 20  to r i g h t .  action at a stage,  The so a  of a s any pathway t h r o u g h t h e  node S, t o t h e s i n k  F.  The  set  management s e g u e n c e s i s t h e s e t o f a l l pathways  of a l l throuqh  Figure  12.  A l t e r n a t i v e  management  sequences  r e p r e s e n t e d as  a  c y c l i c  d i r e c t e d  graph  of  states  and  stages.  118  State  Description w i l d stand clearcut n a t u r a l regeneration, curve 7 p l a n t i n g , curve 9 t h i n n i n g c u t , curve 8 t h i n n i n g cut, curve 5 clearcut  1 2 7 9 3 5  10  F i g u r e 1 1 . A l t e r n a t i v e management sequences r e p r e s e n t e d as a c y c l i c d i r e c t e d graph.  the  graph The  large  from  S ' t o F.  g r a p h i c a l approach  number  of  i s an e f f i c i e n t  alternative  management  number o f pathways t h r o u g h t h e ; g r a p h Starting labelled node.  with  each  pathways back  be  arc  number o f pathways back incident  to S i s the l a b e l  from. The l a b e l s are  sequences.  t h e s o u r c e node S, t h e v e r t i c e s  with the t o t a l  For  may  structure to store a  associated  The  easily  total  counted.  o f t h e graph a r e to  the  source  a t a g i v e n v e r t e x , t h e number o f of the vertex that  with each  summed t o f o r m t h e v e r t e x l a b e l .  arc incident The l a b e l  the a r c  came  at the vertex  computed f o r t h e  119  sink  node F i s t h e t o t a l The  counting  Section the  3.3.2  activity  subscript to  input  to  the  stage states  v  performing incidence  t  v  t <  fc  to the  are  and  i n the  v  can  be  and  fc  MI  pine.  unit  As  be  vertex  of  to a s t a t e with  the  o f as t h e a r c l i n k i n g  v  the .  The  chosen the  two  v :  (4.2-1)  t  inversion Eg.  on  (3.3-19),  cf arcs incident  Then n (F) can  on  i s the s e t of  i  the  dropped  state  M ( v , a ) t  The  are c o n s i d e r i n g  is v . V  (3.3-18)  of  between  MP2.  b e f o r e , the  t corresponds  thought  of  u can  we  i n p u t t o s t a g e t . Each  Eg.  graph.  the r e l a t i o n s h i p  on o u t p u t  function  identity  be  notation  treatment  of Scots  is v  that  state  states  n(v )  =  t  expressed  multistage formulation  beginning of stage  states  the  help e l u c i d a t e  the  unit  transition  discrete  Let  and  be  pathways t h r o u g h  n o t a t i o n , as i t i s u n d e r s t o o d  management a c t i o n  input  will  graph  at the  state  and  treatment  discrete graph  a l g o r i t h m can  corresponding  simplify  a single  number o f  on  Eg.  p r o v i d e s an  n(v ) =  E  recursively  n(v )  =  using  the  expression f o r  the  a given vertex  number o f pathways  be f o u n d  (4.2-1)  E  and  v ;  from  the  source  from  the  relationship  n ( M (v , a ))  node S t o  (4.2-3)  120  for  all v The  (Figure this  V  and t = 1, 2, . ..,  T.  management s e g u e n c e g r a p h 12) h a s been  simple  f o r the demonstration  l a b e l l e d using  problem,  there  this  are  63  procedure. alternative  problem  Even  for  management  seguences. If the  t h e management a c t i o n  arcs,  t h e n t h e MP2 p r o b l e m  seguence F,  i s eguivalent  Formulated  solution optimal  as a  path.  function  Using  in  The  model  the state  manner from  t = 1,  management  action  of the output  t  t  state  are  t  o f f i n d i n g t h e optimum  section  (Eg, 4.2-1) a l l o w s  a forward  R (v ,a )  t o f i n d i n g t h e optimum  network  described  function  returns  1 0  ,  the  3.4,1  inverse  the  the s o l u t i o n procedure  S to  programming  c a n be used of  on  management  pathway from  dynamic  form  placed  to f i n d the transition  t o proceed i n  T. return v  E  with  (v ,a ) the  can  inverse  be  made a  transition  function:  R  " (v ,a ) = R ( M ' ( v , a ) ) t  In  fc  t  t h i s f o r m , t h e management a c t i o n  the (arc)  cost a  fc  For  or  profit  of entering  (4.2-4)  t  return  s t a t e v;  c a n be i n t e r p r e t e d v i a management  as  action  , stage t=1, t h e r e  i s only  one a r c from  S t o v , and  the  A graph d e f i n e s t h e p u r e l y s t r u c t u r a l relationships between the nodes, while a network bears also the guantitative c h a r a c t e r i s t i c s o f t h e nodes and a r c s . F o r a c o n c i s e d e s c r i p t i o n o f network models s e e E l m a q h r a t y (1970). 1  0  121  maximum s t a g e  return i s  l  f  for  a l l  v-^  programming  v .  V  (  For  2  recursion  f (v ) t  t  - V  =  W  ,4.2-5)  s t a g e s t = 2,  ..,, T, t h e u s u a l  dynamic  takes the form:  W V V  +  Maximum  (4.2-6)  where  f  At  the f i n a l *  . .., a  The  simplified  stage  v  )  T  to  in a  f  t-i  ( M  V i ' V  = F, and t h e back t h r o u g h  inversion  tracing  the  optimal  Section  complicated  action.  state  3,3.2, state  pathway  the optimal  o f MP2 of  (4.2-7)  1  i s easily  the  solved  by  as a f u n c t i o n  than  function  would or  c f the output  ,  forward  function  formulation  inversion  transition  a  T  returns.  a r c backwards r a t h e r  state  a ,  stage  transition  However, i n t h e more g e n e r a l  f o r an i n p u t  management  =  formulation  as s t a t e  a stage.  described  model  a  network  recursion,  solving  ( v  c a n be t r a c e d  1  through  t-i t' t  is  forwards of  MP2  involve  simulation state  and  122  4.2.1  Generating  The  The Graph Of A l t e r n a t i v e Management  network  flexible  and  alternative  formulation  efficient  management  alternative  extremely  tedious,  problems,  To make  program  was  structural  o f MP2 i s c o n c e p t u a l l y s i m p l e , y e t  enough  management  to  encompass  seguences. graph  and  Of  course,  adding  the  i f not i m p o s s i b l e , this  approach  written  that  generates  outline  aspects  o f t h e p r o b l e m , t h e c o s t s and can  usually  be  management  assigned  computed  as  during  needed  generating  separate  step, the r e s u l t i n g  calibrated  with  benefit  regenerating  the  during  Through  cost  a  arc  structural  and  is  feasible, encoding  later  aspects  solution  a the  from  defined  generation  the  network  a  a  The q u a n t i t a t i v e  profits  on  the  s t e p , but a r e solution  step.  of the problem as a  for different  at  of  returns  seguences,  management s e q u e n c e  returns  assumptions  arc  graph  problem.  the  sets  formulating  computationally  simple  arcs,  the  large  f o r any b u t t h e s i m p l e s t  c h a r a c t e r i s t i c s o f t h e management of  Seguences  graph  treatment  time,  f o r each  can  be  units or  rather  than  variation  on t h e  storing  a  followed  by t h e g r a p h g e n e r a t i n g  problem. The described 1 :  procedure  program  is  below:  Read  in  and  generating time  set  program  frame  for  up  allows the  represent  a different  is  be  to  disturbance  t h e problem a user  problem,  to so  time  frame.  define that  The g r a p h  a  variable  each  stage can  time span. F o r example, i f a  naturally f o r 40 y e a r s ,  regenerated a  single  and stage  grown of  stand without  40  years  12 3  could  cover  represent  this  2-year  intervals,  resolution  for  completely  flexible  effort of  t i m e s p a n . The f o l l o w i n g  scheduling  c a n be c o n c e n t r a t e d  planning  parameter  list  each s t a t e , the  be -  than  action. problem  This  solving  time p e r i o d s t h a t a r e  being evenly  for a  consists  defining  allocated  treatment  of  the  a  over  unit.  description  treatment  i n simulating  t h e graph  earliest  unit  the state  description  and l a t e s t t i m e  The  o f the and a  simulation  t r a n s i t i o n s . For  consists of:  period  that  the s t a t e can  entered, t h e minimum  transition - the the  and  into  costs  state  maximum  per unit  area o r per u n i t  (revenues a r e t r e a t e d  t h e wood volume p r o d u c e d  -  a flag  simulating  from  (from and  i s  years  of  the  to  be  volume o f  on e n t e r i n g  generated  at  entering  costs),  the s t a t e , cost  or  solution  revenue t i m e by  the t r a n s i t i o n i n t o the s t a t e ,  of output  the current  Generate  in  as n e g a t i v e  t o s i g n i f y whether a d d i t i o n a l  information  a list  duration  the s t a t e ,  -  -  means t h a t  sufficient  a s p e c t s o f t h e management s e g u e n c e g r a p h  model t o be used  -  thinning  might  horizon.  outline  structural  provide  i n those  Read i n a management o u t l i n e management  a  t i m e frame  most i n t e r e s t , r a t h e r  the  to  stages  the  states  graph,  into  may  be  entered  directly  state. Arcs  stage, from s t a t e , sorted  that  are generated as t h e 4-tuple  at stage,  ascending  order  at state) by s t a g e  (Figure  13)  and s t a t e t o  124  facilitate 4  :  The  t h e dynamic  a r c s a r e added  programming  to a l i s t .  I f t h e r e a r e more t r e a t m e n t  'from' states  output states  t1  t  +  ' from' stage F i g u r e 13.  5 : A  step  step  The  list  Otherwise  i s s a v e d on a permanent  problem  investigation to  2 through 4 are repeated.  generating  file.  was  constructed  similar  of thinning  strategies.  Stands of Scots pine  be  50  years  to Kilkki's  20 y e a r s .  The  to  intensity  o f up t o t h r e e  followed stand  by a c l e a r c u t .  density  together  the t i m i n g  with  levels a  (1970)  o l d , grown w i t h o u t i n t e r v e n t i o n  s t a n d s o f 6 cu.m./ha. At age schedule  proceed  5.  graph  test  assumed  'at' stage  L a b e l l i n g o f a r c s produced by t h e graph program.  units, to  solution process.  and The  problem  at  age  precedence  c o m b i n a t i o n s o f t h i n n i n g s and  states,  20, graph  are  decision  defined listed  showing  a clearcut.  problem  were from was  thinnings,  i n terms  of  i n Figure  14,  the  possible  125  c c  State 1 2 3 4 5 6 7 8 9 10 11  Figure  14.  —-  Description DL = .5 cu.m./ha. a t age .75 " 1.00 " 1.50 " 2.00 " 2.50 "3.00 " 4.00 " 6.00 clearcut i n i t i a l (dummy) s t a t e  20  S t a t e d e f i n i t i o n and precedence graph o f a m u l t i p l e t h i n n i n g d e c i s i o n problem ( K i l l k i ' s model).  126  Quite  complex  precedence  g r a p h . From t h e  possible  to  (DL=2).  Second  states  4,  states  3,  from  states  2 or  can  result  be  to  In of  the  1.  1.  addition state  between  year  3 at  are  the  from  .75,  state  .5,  state 7 and  6  5 to  respectively). result  in  S i m i l a r l y , subseguent t h i n n i n g s  from  state  4  The  1,  second t h i n n i n g  and  are  only  2 or  a  (DL=2.5),  1.0,  in  (Dl=6), t h i n n i n g s  possible  (Dl=1,5, 5,  6  encoded  can  but  a thinning  clearcut,  to t h e  initial  state  state  10,  from  state  3 can  i s accessible  precedences,  51  and  by  thinning  states  year  Any  year  71. 86.  63,  state  2 at  may  not  occur  before  information  generate  (7, 6,  and  subseguent  State  year  clearcut  only  from a l l  4 may  year  65,  be  and  y e a r 76  5)  the may  timing  activity be  entered  thinnings  must  be  entered  year  61,  state  but  on  at  1 at  must be  year  done  67.  before  91. The  data the  state  9  t r a n s i t i o n s i s necessary to  accomplished  year  1  state  are  states.  The  A  and  in state  graph.  state  (DL=3) ,  state  state  thinning  7  strategies  initial  thinnings  2,  However, 3,  thinning  program e x e c u t i o n command and  set  are  listed  graph c o n s i s t e d  second  stage of  duration. first  As  50  an  initial  years.  of  resolution  The  management  time  stage of  outline  frame d e f i n e d  1 year f o l l o w e d  S u b s e q u e n t s t a g e s were a l l o f were t o  the  stands'  and  solution  take  growth, effort  during  2  place  this  a  years  in  time the  by  for  the frame  period  of  management.  After program  of  VII.  management a c t i o n s  years  concentrated intensive  no  51  i n Appendix  the  generating  was  used  to  the  graph t o  display  a permanent  information  about  file, the  a  separate  graph.  Figure  127  15  i s a management o u t l i n e r e p o r t  listing  o f t h e management o u t l i n e d a t a  1 M M M M 1  M i l  !  J  :  E N T R Y STATE FIRST LAST 15 | 9 ! 1 1 6 I 5 I 4 i 3 } 2 ! 1 I 10 J 10 !  41 53 —• w  53 61 63 65 67 76 93  41 71 71 71 86 86 86 86 91  PINE,  AGE  |  j  | | J | | | | | | I  | | | | \ | | | | I  NETWORK  and  of the appropriate  duration  a  set,described  VOLUME CUT I I I I I I I I 1 1 1  and t h e g r a p h  stage  simply  formatted above.  SIM OUTPUT STATES | | | | I | | | J 1 |  HAS 29886 F E A S I B L E  maximum  may be s p e c i f i e d ,  i s  50  15. A management o u t l i n e r e p o r t Scots pine.  Minimum state  SCOTS  DURATION COSTS MIN MAX AREA VOL  THE CORRESPONDING SEQUENCES  Figure  and  F T T T T T T T T T F  ! 9 ! 7 6 5 10 4 ! 3 2 1 4 3 2 1 10 ! 3 2 ! 1 10 ! 2 1 10 1 10 | | 10 | 10 | |  MANAGEMENT  f o rmultiple  times  of t r a n s i t i o n s  program  length. In the  thinnings of  will  generate  demonstration  option  i s n o t used  occur  during  one t i m e s t a g e  (two  (SIM  FLAG)  set  (T) f o r any s t a t e s i g n i f i e s  transition  into  the  arcs  problem,  this  to  into a  and i t i s assumed t h a t a l l t r a n s i t i o n s  true state  years).  i s to  be  The  simulation  simulated  flag  that the  with  the  128  appropriate The Section  model, t o  provide  report  program  4.2.1  and  The  of  report  graph  the  (e.g.  graph  12),  i n the  demonstration  also  in  Figure  9 to  16.  state  through  The  the  graph  stage  but  Finding  The  program  The  process the The  the  dense t o  generated  'no  correctly  that  be  the  activity  included  the  here.  from  the  number o f  in this  logical  represented  (e.g.  represents  VII.  p l o t of  graph i s  apparent  partial  check f o r  change' a r c s  Management  dynamic  A  g r a p h . However, t h e too  (feasible  i n Appendix of  used t o  c o r r e c t l y solved  program the  plot  in  state  pathway arcs.  The  form.  Sequence  solves  the  programming  network  algorithm  problem  described  in  4.2,1.  The arcs.  follows  be  Optimum  computer  generally Section  The  be  decreases the  network problem cannot  4.2.2  was  2)  described  graph.  graph i s i n c l u d e d  a d d i t i o n of  9 during  algorithm  activity  generate a  of  information.  29886 u n i g u e pathways  which can  problem  benefit  counting  generated  A s i m p l i f i e d version  and  the  i n the  will  Figure  inconsistencies  through  arcs  program  uses the  calculated  management s e g u e n c e s ) listing  cost  arcs arcs  SOLVE r e a d s i n t h e are  p r o g r a m , has  so  that  s e q u e n t i a l l y i n the  computational  relationships  ordered  that  burden  of  to  file  the arc  list  of  solution algorithm  can  list,  dealing  would n o r m a l l y  been t r a n s f e r r e d  graph  the  be  as  on  a single  with  associated  graph  a  pass.  precedence with  generating  a  network program.  Figure  16.  A  graph  problem  representing i n  Scots  the  pine  a l t e r n a t i v e  management  sequences  of  a  m u l t i p l e  t h i n n i n g  130  Consequently the for  the  s o l u t i o n p r o c e s s i s f a s t and  test  problem  involving  inexpensive  29,886 a l t e r n a t i v e  ($1,19  management  sequences) , The  solution  program  transition  represented  reguired  provide c o s t / b e n e f i t  to  appropriate return be  on  treatment  the  entered  site  VII.  with  unit  disccunt  the  the  an  function  specify  program  reports  problem.  The  discount and  rate  72-74.  82-84 the  and  The tested  output  maximize  included  displayed of  volumes  A final  PNw)  same g r a p h  can  such  as  were  cut  was  cu.m./ha. The  for  entering  sequence  is  box  discount solution the  worth  and  on  a  i n Figure  during  state.  interior  with  a  54-56  cu.m./ha.,  during  Figure  test  17.  years  158  unit  select  the  scheduled  a given an  treatment  present  129  Appendix  VII,for  graphically net  in  , assign  were s c h e d u l e d  cut  harvest  193  the the  reports,  Appendix  maximizing  thinnings  management  17  Note  years  represents that  optimum, w i t h  the no  'tight'.  optimality with  of  the  in  yielded  constraints  nature  assign  are  2%,  is  estimate  i s included  l a g . Examples o f  time c o n s t r a i n t s  optimum  problem  a regeneration  The  respectively.  to  model p a r a m e t e r s  user t o  (e.g.  objective of  test  number and  r e s u l t s are  with the  the  I f necessary,  i s executed  of  of  checks i f a s i m u l a t i o n  information.  model  a l l o w s the  model, c o n t r o l  and  and  nature  rate.  for  A c o n t r o l card  rate,  arc,  different settings  runstream  objective  an  the  s t a t e t r a n s i t i o n . Consequently, the  index or A  by  interprets  the  (SIMOPT) d e s c r i b e d  of  the  management s e g u e n c e  simulation i n Section  optimization 4.1.  was  subseguently  supervisor  A policy f i l e  program  containing  the  131  F i g u r e 17.  o p t i m a l management sequence c a l c u l a t e d by embedding K i l k k i ' s Scots p i n e model i n a network f o r m u l a t i o n .  n  132  management s e g u e n c e s e l e c t e d created the to  program  w i t h t h e management a c t i o n s o c c u r r i n g a t t h e m i d p o i n t  time occur  s t a g e s . F o r example, t h e f i r s t i n the stage  simulation to  by t h e network s o l u t i o n  occur  simulated  corresponding  optimization  policy,  at t h e midpoint and o p t i m i z e d  to  t h i n n i n g was years  the f i r s t  of the s t a g e , i n the user  54-56.  7 PARAMETERS  For  AGE  VOLUME BEFORE CUT  55 73 83  279.OC 298. 00 193.00  PRESENT NET  129.00 158.00 193.00  WORTH :  OBJECTIVE VALUE IS ? OPTIMIZE # 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16  VOLUME REMOVED  session recorded  5 VARIABLES NET RETURN 813.94 1198.79 1661.25  1071.89 1071.89  20  Z* 1071.891 1072.906 1072.906 1072.906 1072.906 1073.575 1073.575 1073.57 5 1074. 574 1074.574 1074. 574 1074.574 107 4. 574 107 4.574 1074.998 1074.998 1074.998  V A R I A B L E S 1 3 11 73. 00 83.00 55. 0 0 54.50 73.00 83.00 54.50 73. 00 83. 00 54.5 0 73.00 83. 00 73. 00 83. 00 54.50 54. 23 72. 89 84.60 54.23 84. 60 72. 89 54.23 72. 89 84. 60 53.93 73. 15 83. 59 83.59 73. 15 53. 93 53. 93 73. 15 83. 59 53.93 83. 59 73. 15 83. 59 53. 93 73. 15 73. 15 83.59 53. 93 53. 94 73. 17 83. 54 53.94 83.54 73. 17 53. 94 73. 17 83. 54  the  t h i n n i n g was assumed  55 y e a r s . The p o l i c y  ? SIMULATE  of  scheduled  below;  ? READ KKDF 4 POLICY KKDP ,  was  ARE  READ IN.  was  133  17 18 19 20  1075.108 1075. 108 1075.108 1075.108  53.93 53. 93 53.93 53.93  73.16 73. 16 73.16 73.16  83.53 83. 53 83.53 83.53  AFTER 20 ITERATIONS,THE BEST RETURN IS 1075.108 FREE VARIAELES : 53. 93 73. 16 83. 53 ? STOP  When present  SIMULATEd  through  worth s l i g h t l y  solution small  program  in  OPTIMIZATION command the  the  policy  that returned  the  refines  present  by  computational the  net  convergence t o  the  same  optimality  the  policy  of  than  ,  has a n e t  the  network  ($107-1.89 v s . $1072. 3) . T h i s i s p r o b a b l y  differences  increases  less  SIHOPT  worth  policy  approach t o o p t i m i z a t i o n .  policy  or  tc  does  procedure.  The  slightly,  and  $1075.11. not  confirm  However,  only  prove  Of  course,  the  the v a l i d i t y  consistency  due t o  global  of e i t h e r  i s , at  least,  encouraging, A the  last  e x e r c i s e attempted  demonstration  of the s e n s i t i v i t y  changes  in  Kilkki's  model, v e r y  decision  problem,  outline appendix.  the  and The  disccunt  As  would  one  scheduled 1%  2%,  expect,  i s c l e a r c u t a t 66  optimal  test  i n Appendix are  management . . . . . .  8%  the second  thinning years.  policy  V I I . The  management in  the  corresponding  are recorded  on  Figure  t h i n n i n g and h a r v e s t  i s eliminated  to  thinning  included  seguences  rate  was  problem i n v o l v i n g multiple  also  a t y o u n g e r ages as t h e d i s c o u n t  and 8%, t h e s e c o n d  stand  of the  to the previous  runstream  3%,  t h e network f o r m u l a t i o n  A second  i s described  optimal  rates of  rate,  similar  solution  discount  with  rises.  to 18.  cuts are At  completely  rates and t h e  134 CD  40.0  3 0 . 0  5 0 . 0  6 0 . 0  Age  Figure  18.  S e n s i t i v i t y discount  of  rates(  the  20.0  7 0 . 0  S O . O  100.0  (years)  optimal  K i l k k i ' s  management  model  i n  the  sequence network  to  d i f f e r e n t  formulation).  135  4.2.3  L i m i t a t i o n s Of  The  naturally  treatment was  unit  state in  such  a list  age  is  into  area dbh  1 1  state  incorporated  transformed approach  specify  crown  the  resulting  gross s i m p l i f i c a t i o n Goulding's  model  diameter  takes  a  discrete  be  states.  c l a s s e s that Goulding  together express  the  the  with  the  state  in a discrete  stand  i n increments on  as  average  basal area  few  of as  states i s prohibitively  of the in  of  to  One  stand  proportion of  discrete  distribution  average  c l a s s i s expressed  set of  dynamic  the  five  diameter  set  the  stand  c l a s s e s d e f i n e d about  if  stand  dbh  of  age  basal  However,  average  stand  stand  might a d e q u a t e l y  diameter  model i s t h e  the  diameter,  each  described  proportions of  thinnings  the  levels.  multidimensional  definition  the  t o r e c o r d the  For example,  density  a  model,  1  a t b r e a s t h e i g h t . The  the  acceptable  of  Kilkki s  D o u g l a s - f i r model. As  diameters into  i n a number o f dbh .  discrete  of Goulding's  leaving an  would be  by  by  however, f o r models w i t h  of sample stem  stage,  (cu.m./ha.)  p i n e , as s i m u l a t e d  as G o u l d i n g ' s  programming  Formulation  state variable  represented  S e c t i o n 4.1.4, t h e  and  to  Scots  arise,  spaces,  Discrete  continuous  of  adequately  Complications  The  state  definition,  a  network  problem  is  stand form.  falling  in  and  the  .1, 10  uses  values,  large.  the  Without  formulation  of  computationally  infeasible. Consider  the  following  compromise  approach.  The  large  T h i s r e p r e s e n t a t i o n o f t h e s t a t e o f a s t a n d has been used a growth and y i e l d model f o r M o n t e r e y p i n e ( P i n u s r a d i a t a ) , C l u t t e r and A l l i s o n (1974). 1 1  in by  136  number  of  states  that  For  possible record  example,  stand  after  the  first  network  only  state  i t has time.  optimal  to  distribution  i s not  but  to  implications reference  state 5  solution only  that used  of t h i s  Consider  previous  by  state  variable  been t h i n n e d  sufficient schedule The  knowledge  a second  optimum  level  s t a t e 3,  it  is  the  final  1 2  .  to  conceivable optimum  that  area,  for  above,  the  to  from  e  e  dbh  stage, The  described  (first  (a  with  thinning)  clearcut).  relevant  to  At the  transition.  The  that  the  selecting  25%  of  the  information  its  basal  t o base t h e  a second  pathway i f t h e  to t h e  i n c l u d e the low  the  is  not  density  to  clearcut. low  density  planting cost.  t h i n n i n g , s t a t e 5, initial  that  d e c i s i o n whether  proceed d i r e c t l y  associated  area  might were  i2 wagner (1969, p. 343) f o r a c o n c i s e description c o n d i t i o n s and a s s u m p t i o n s o f m u l t i s t a g e a n a l y s i s . s  right.  Markov a s s u m p t i o n  s t a t e 4 might  because of the  the  d e c i s i o n v a r i a b l e s i s summarized  which  t h i n n i n g or  pathway  of s t a t e 4  Obviously,  once by on  the  left  easily  stages  current  a  or s t a t e 6  b a s e d on  f o r the  basal  s t a t e t o any  for  most  cut  about  c o n d i t i o n of  node i s s t o r e d . The  return  transitions  values  has  the  by  occurred.  resulting  input/output  19a.  optimal  stand  an  Figure  is  cf i t s  distribution  (a s e c o n d t h i n n i n g )  information  25%  the  manner, from  approach are  algorithm  the  by  as  state-stage  as  example, the  defined  has  network s o l u t i o n d e s c r i b e d  dbh  compute  t o an  be  i n a forward  s t a t e , the  states i s replaced  a management a c t i o n  been t h i n n e d With the  pathway  only  that  distribution  4 might  i s processed  each a c t i v i t y  into  diameter  Yet be  on  higher  of  the  ,137 State 1 2 3 4 5 6  Figure  Description i n i t i a l (dummy) s t a t e 1200 stems / a c r e a t age 20 y e a r s 600 stems / a c r e a t age 20 years t h i n n i n g s t a t e , removed 25% o f s t a n d b a s a l t h i n n i n g s t a t e , removed.-'15% o f s t a n d b a s a l clearcut  19.  area area  Goulding's model i n a network d e c i s i o n problem, w i t h a s i m p l i f i e d d e f i n i t i o n o f s t a t e ( a ) , and r e s t r u c t u r e d t o a v o i d non-Markov s i t u a t i o n s ( b ) .  138 (state  2).  The  information  definition  about  between d e n s i t y  the  of  previous  costs  and  state  4  carries  too  t r a n s i t i o n s to a l l o w  profits  from  a second  little  the  tradeoff  thinning  to  be  evaluated. These  situations  restructuring in  the  the  are  often  problem, a l t h o u g h  solution  pathways  can  process.  separated  by  adding  the  5'  4  and  5  results  separate  levels  interpreted 2,  1200  have been  respectively.  in  density  and  as  stems  a thinning  of  per  at  area  20.  The  transitions  4-5*  can  be  On  selecting  tradeoff  between s t a n d  thinning  is  density  restructuring  f o r c e s the  s o l u t i c n program  sequences  that  differ  programming  determine the fin  timing  example  of  stocking  the  state  of  20  of 5  years  in  is the  Markov  state  and  be  state  and  the  into  the  can  1  4,  to  states  state  state  utilizing  of  i n the  is  and  the  problem  t o enumerate and  6,  the  the  second  this  manner  in  t o two  management  included. used  only  The to  actions. Appendix  network f o r m u l a t i o n up  compare  advantages are  management  provided  in  management a c t i o n s  decomposition  G o u l d i n g ' s model i n t h e levels  a t age  each  previous  transition  19b,  evaluated.  Essentially,  dynamic  the  two  a f t e r being  definition evaluated  the  to  basal  the  the  in Figure  of  15%  in  assumptions.  addition  that  in definition  ( 2 - 4 ' - 5 - 6 ) . The  recognized 4-6  example,  and  implicitly  necessary,  s t a t e s , so  corresponding  age  through  efficiency  whenever  added, i d e n t i c a l  pathways  (2-4-5'-6)  or  For  However,  acre  much l o s s of  redundant  holds i m p l i c i t l y .  4'  with  circumvented  Essentially,  Markov a s s u m p t i o n states  be  thinnings.  to  VIII  that  consider  uses three  139  4.2.4  Summary  The  Discussion  greatest  solving  HP2,  management large  And  disadvantage  the  decision  seguence  f o r a treatment  multidimensional  state  g r o w t h model. The  network  quite  with  variable  essentially,  solution  the  advantages of the efficiency  of  distribution fast  dynamic  and  computer  functions  in  models  that  multistage  but was  pine  of the  in  stand.  should  Section major  suitable  f o r o p t i m i z i n g MP2  automatic  process, 5.  the  Conseguently,  state  However, t h e due  to  the  and  the  structure  used  4.1.6  be  state  the  problems  be  to  restrictions  where  as  of  of  state  transition have for  advantage  f o r m u l a t i o n i s t h a t i t c o n v e r g e s t o an intervention.  and  graph  from  stand  (a s i n g l e  decomposition  to  o p t i m i z a t i o n . The  continuous,  the s i m p l e s t  inexpensive,  Apart are  prohibitively  the  model  to  optimal  demonstrated  formulations  recommended  without  Section  i s the  generated  stand  network  unit,  programming  dimension,  to direct  the  D o u g l a s - f i r model,  management s e q u e n c e s .  subjected  finding  any  Scots  alternative  characteristics  of  required simplification  is  the  approach  approximate  of  Kilkki's  dbh  procedure  network  formulation  when a p p l i e d t o G o u l d i n g ' s is,  to  space  - v o l u m e ) , but  the  problem  set of s t a t e s necessary  effective  of  acceptable  the models  of  the  policy  the network f o r m u l a t i o n i s  as a submodel under  decomposition  control  model f c be  of  an  described i n  140  5 . Joint  Of «nM find MP2  Optimization  In  Section  programming commodity  3.4.3  i t  was  allocation of  problem.  management  t o MP2, t h e  actions  programming  problem  to  subproblems. motivated the  in  Lagrange  subproblem. used  to  In t h i s  of t h e problem.  A synthesis  generated  by  optimization  Dantzig-Wolfe  described,  the of  of  LP  one  master or  more  decomposition,  was  3.4.4 t h r o u g h an a n a l y s i s o f t h e r o l e o f i n the discrete  maximum  form  of  the  chapter, Dantzig-Wolfe decomposition w i l l  the  Timber  RAM  LP  model  with  the  be  network  o f the subproblem.  Program  linear  constrained treatment  the  approach,  multipliers  The l i n e a r  The  multipliers  Section  link  formulation  5.1  This  on a t r e a t m e n t u n i t , s h o u l d  m u l t i s t a g e d e c o m p o s i t i o n was  coordinate  linear  Conversely, a solution  linear  Lagrange  a  o f MP1, t h e  the m u l t i s t a g e s t r u c t u r e with  that  formulation  exploit  the  established  model was t h e most e f f i c i e n t  scheduling  using  V i a Decomposition  Master  approximation  commodities  units  to  to  was s e l e c t e d  t o form  to  i t s extensive operational  unit.  use by t h e U.S. The Timber  creates  a  allocates  certain  sequences  fora l l  For t h i s  t h e b a s i s o f t h e MP1  i n t h e system.  a m a t r i x g e n e r a t o r , which  MP 1  management  a c r o s s a management  BAM  BCFS i n t e r e s t  Problem  RAM  linear  study.  Timber  linear  model,  due  Forest  S e r v i c e and  software consists of model  in  a  form  141  solvable writer  by  that  graphs.  a  commercially  t r a n s l a t e s the  The  {Nazareth,  structure 1S 71)  decomposition  model  closely  that  fellow  Timber planning  BAM  only  of  of  variables  10  linear  those  will  be  c o d e , and useful  model  1 and  here.  The  be  report and  documented  relevant  to  the  notation  will  3.4.3.  time i n c r e m e n t s of  d e c a d e s was  a  tables  i s well  elements  reviewed  Sections  must  LP  solution into the  a l l o c a t e s on  horizon  Two  LP  of  and  available  1 d e c a d e , and  used t h r o u g h o u t  defined.  Let  the  be  x  a  study.  the  area  of  uk  treatment a  (k)  u  unit u  . The The  and  total  treatment total  volume c u t  values  limited  that  by  a  unit that  area  of  the  ,  The  managed by  the  i n decade  t h e s e two set  j  variables  i s managed  i s , of  can  t a k e on  eguations. course,  are  The  related  area  limited  to  of  a  the  unit:  u  r e l a t i o n s h i p between  and  x  C  i s established  through  j  uk  the  seguence  j is C .  cf c o n s t r a i n t  treatment  uk —  a l t e r n a t i v e management  equality  Z  E x  u  k  uk  J2_2 gx , uk  " C.  =0  -i  j=1,  J  (5.1-2)  J  3C.  where  —J-  i s the  Volume f l o w  volume per  constraints  volumes between t i m e  periods.  unit  limit The  area the  harvested  rate of  constraint  i n decade j .  change o f used  harvest  throughout  142  this  study  within  was  C  0  <  j  harvested  harvested  i n period  i n period  l.ic. ,  j-1,  * =  9C  cut i n the present  1  be  i.e.  (5.1-4)  (5.1-3)  J  decade.)  I E o b j e c t i v e was t o maximize t h e p r e s e n t  management  j must  j=1, . .., J  ±- °- j-l  i s t h e volume The  the  t h e volume  ± 10% o f t h e volume  c.  (C  that  net  worth  of  unit:  u K Maximize  3R  where  —  3x  treatment  is  R  =  Z  Z  u  the  k  x  (5.1-5)  — u k 3x , uk  present  net  worth  of a unit  c f area of  uk  unit  u, when  managed  a  by a l t e r n a t i v e 3C  The commodity  input-output  coefficients  — 3 — 9x  a  coefficient  R  are organized  ox  • U  a s uxK v e c t o r s :  and  price  , uk  uk  {  > 3x  Additional dimension the  uk  1 3x .  structural  uk  /  2 3x ,  , ...  ,  J  uk  coefficients  s e g u e n c e i s known  }  (5.1-6)  uk  usually  c f t h e v e c t o r t o > J+1 e l e m e n t s .  management  , ...  3x ,  This  increase  the  expression  as t h e LP v e c t o r o r a c t i v i t y .  of  143  5.2  Decomposition  Each feasible  Cf The  iteration allocation  variables  at  linear  of the of the  each  the  marginal  and  values  interpreted  resulting The  from  decade j i s the  are  to the  dual  marginal  allocation.  variable  In  cf the  commodity  in a  The  dual  values of  section  'constrained  p e r t u r b a t i o n s of the  value of the  results  1 3  commodities.  the  r a t e o f change  feasible  algorithm  constrained  were d e s c r i b e d as  as t h e  marginal  LP s i m p l e x  iteration  commodities corresponding  Model  the  3.4.3,  derivatives,  1  objective function commodities.  volume  a s s o c i a t e d with  harvested  in  Eg. ( 5 . 1 - 2 ) ,  , <5 C , D  j=1,...,J, prices  are  In  the  decomposition  incorporated into  the  algorithm,  these  commodity  o b j e c t i v e f u n c t i o n of  the  MP2  u, a t  time  s u b p r o b l e m . Eg. ( 3 . 3 - 2 0 ) .  R  The t,  u  T Z R . (v . ,a ) . ut ut ur  =  amount c f commodity by  management a c t i o n  j produced a  , is c ut  its  marginal  v a l u e , i s added  R  u  T £  = t  =  1  on  treatment  . The  unit  commodity  weighted  by  jut  to the  ( R + Z c , ut . ]ut  r e t u r n f u n c t i o n of  9R  .  MP2:  -1\  (5.-2  f 5  2  D  The s i m p l e x a l g o r i t h m i s t h e LP s o l u t i o n p r o c e s s devised by George Dantzig and s h o u l d not be c o n f u s e d w i t h t h e s e g u e n t i a l s i m p l e x s e a r c h t e c h n i q u e d e s c r i b e d i n S e c t i o n 4.1,1.  144  When t h e sequence a  management  k,  the  actions  return  per  form  unit area  alternative of  treatment  management  k  managed  by  (k) . is  u  6R 6 X  T  uk  t=l  Note t h a t t h i s equivalent  t o Eg.  maximizes the in  terms  LP  master  master  ut  form  j uk 8 X  of  the  ( 3 . 4 - 1 8 ) , and  the  6  LP  master  j  C  MP2  objective  problem, at  is  it  management  problem.  buys/sells  the  solved  marginal to  function  t h a t s o l u t i o n of the  In  the value.  maximize t h e  alternative when  the  in  the  i s scarce  commodity  is  subproblem  effect,  c o n s u m e s / p r o d u c e s a commodity t h a t  problem  subproblem  3 X  d e c i s i o n d e r i v a t i v e of the  of  subproblem  3c.  9R  from/to  the  Consequently,  return  net  of  the  commodity  values. The briefly 1  :  decomposition  Assume  that :  used  in  this  study  can  be  stated.. that the  a candidate  2  algorithm  IP  has  been p r o v i d e d  with  s e t o f management s e g u e n c e s a c o n s t r u c t e d  i t contains  Optimize  master problem  the  at l e a s t  linear  one  feasible  model w i t h  the  solution  so  a . f  current candidate  set  a. 3  :  Obtain  the  commodities,  marginal -—  (j=1#  values  of  the  constrained  J).  j 4  :  For each MP2  using  treatment the  u n i t , solve the  network f o r m u l a t i o n  augmented o b j e c t i v e f u n c t i o n , i . e . , f i n d  of * a u  145  such 5 :  that  Create  Eg. (5.2-2) i s m a x i m i z e d .  LP  v e c t o r s from  activities 6 :  Optimize a.  5.2.1  (IBM,  LP s o l u t i o n  MPSX  program  control  as  and  the  the  FORTRAN  demonstrated  i s listed  parameters  facility  is  (matrix) file.  (IBM,  subsystems is  convergence file  commodities is  then  to  used  i s  o f f a line i s then  an  and  set  optimum, the  are selected  transferred  file up  invoked  under which  READCOMM  d a t a between t h e  t o MPSX, t h e LP  model  and l o a d e d o n t o t h e p r o b l e m  in  core  values  and l o a d e d i n t o  t o t h e FORTRAN  and t h e  starts  The MPSX  transfer  the current  marginal  i s  and f l a g s . to  system  4.2 . A f l o w c h a r t provided  SUBIN  (MPSX)  solution  IX. The program  and MPSX. On r e t u r n i n g  read  The p r o b l e m  problem  1971b)  network  procedure  algorithm  System E x t e n d e d  i n Section  i n Appendix  o f MPSX. A FORTRAN  decomposition  program  r e a d s i n v a r i o u s problem  FORTRAN  continue  MPSX  Programming  20) o f t h e d e c o m p o s i t i o n  program  With  implementing  t h e IBM M a t h e m a t i c a l  described  (Figure  solution,  i soptimal.  Decomposition  1971a) a s w e l l  SOLVE  i n an i m p r o v e d  set  s t e p 3.  Dantzig-Wolfe  utilizes  model w i t h t h e c u r r e n t c a n d i d a t e  step results  The c u r r e n t  The  (u=1, ..., U) and add t h e new  u  the c u r r e n t a.  the l i n e a r  If this  with 7 :  into  a  and basis of  optimized. i s saved  the  on t h e  constrained  MPSX v a r i a b l e s .  procedure  On  Control  SUBDC which  loads  146  FORTRAN  MPSX  START  i LP F i l e  SUBIN'  (matrix)  Problem oplions and /parameters.  CONVERT  0  Graph File  SOLVE Network Solu t i o n prog.  Generate LP V e c t o r s  * _ user s u p p l i e d MPSX procedures r e g u l a r MPSX procedure are u n d e r l i n e d  Output Procedures F i g u r e 2 0 . F l o w c h a r t o f t h e MPSX Dantzig-Wolfe decomposition program.  147  and  transfers  control  Under c o n t r o l off the  the  o f program  GRAPH f i l e ,  commodities  network  subproblems  file  are  problem  from  MPSX  solved.  a s Timber  added  i s then  invoked, the  o f an MPSX REVISE  did  i s again  5.3  An Example P r o b l e m  this  vectors  from  optical  transferred  section,  approach  appropriate  problem  file.  with  for a  Kilkki's  Goulding's thesis  Sects  i s guite r e a l i s t i c  REVISE  The r e v i s e d  procedures are  I f improvement i n  subsystem.  pine. A  pine  involving  D o u g l a s - f i r model  model  in size  Approach  demonstration  written  and i n c l u d e s a more c o m p l e t e e c o n o m i c  t h e example  the  i s described  the decomposition  optimize  in  To D e m o n s t r a t e The D e c o m p o s i t i o n  an example p r o b l e m  a  values are selected  t o t h e SOLVE  of  However,  management  C o n t r o l i s then  and o u t p u t  management o f 85,000 h e c t a r e s o f S e c t s  Columbia.  and t h e  the o p t i m i z a t i o n , the  the  be more  values of  b a s i s r e s t o r e d , and  o c c u r , t h e new m a r g i n a l  control  read  i s unloaded.  by t e r m i n a t i o n o f e x e c u t i o n .  and  would  file.  t h e model on t h e problem  i s considered  followed  objective  In  optimal  s e t up, t h e p r e v i o u s o p t i m a l  solution  models a r e  RAM LP v e c t o r s , a r e w r i t t e n t o  o p t i m i z e d . I f no improvement r e s u l t s current  The  o f MPSX, t h e new LP to  SOLVE.  v i a READCOMM,  back t o MPSX and t h e SOLVE system  Under c o n t r o l file  SOLVE t h e network  are  i n t h e format  transferred  subsystem  and s e t up i n c o r e . The m a r g i n a l  are obtained  seguences, expressed line  t o t h e network s o l u t i o n  in  British  i s cheaper t o  model.  Otherwise,  and c o m p l e x i t y .  148  5.3.1  Problem D e s c r i p t i o n  The Scots  management  pine.  consists of eight  i n Appendix  X. The g r o w t h  curves  10.  Treatment  is  of  management o u t l i n e r e p o r t i s c o n t a i n e d represented  unit  units  below and t h e  are those  treatment  treatment  described  T01  Each  unit  i n Figure  Unit  Treatment timber, value  unit  1 (T01) i s 15,000 h e c t a r e s  with  of the  $/cu.m.  The  an  of  overmature  a v e r a g e o f 270 cu.m./ha. The  timber, timber  net is  of  to  harvesting  estimated  costs,  be h a r v e s t e d  is  i n t h e next  5 20  years. Three r e g e n e r a t i o n High the  density stands  density  8)  regeneration  costing  $10 $/ha.  cuts  Treatment  unit  similar  to  Consequently, management The  on may  (curve  the be  2 T01,  9, w h i l e  results  Natural  are  to  be  considered.  a t a c o s t o f 30 $/ha. w i l l  cn growth c u r v e  Depending  T02  planting  (curve  thinning  options  in  planting to  a  of  a  lower  l o w e r c o s t , 24 $/ha.  7) r e g u i r e s s i t e  type  place  preparation  regeneration,  up t o 3  scheduled.  (T02)  is  except  5,000 that  hectares i t  the t r a n s p o r t a t i o n c o s t  is  of  timber  more  component  remote. of  any  a c t i o n i s higher. unit  will  yield  $/cu.m. The r e g e n e r a t i o n  310 cu.m./ha. w i t h options  a value  are i d e n t i c a l  to  of 5 those  149  of  T 0 1 ; high, d e n s i t y p l a n t i n g , low d e n s i t y  natural  regeneraicn.  increase while  site  preparation  c o s t s a r e unchanged  thinning  costs  identical  in  but i s  3  currently  4  T04.  scheduled  before  Treatment  unit  5  (curve  9 ) . There  Up  three  thinning  to  10  are cuts  that  20,000 may be  density  o l d . Up t o t h r e e  the harvest  cut  as  T04,  thinning  on  the  cuts  15,000  6 (T06) i s c u r r e n t l y 50 y e a r s (curve  unit  7  8).  times  (107),  The before  with  t o TC6 e x c e p t  10,000  that  hectares  the f i n a l  an a r e a  o l d and o f be  harvest cut.  o f 10,000  harvesting  may  hectares,  costs  are  2  higher.  Treatment is  density  before  up t o t h r e e  identical  $/cu.m.  o f age 20 y e a r s ,  (T05) i s t h e same h i g h  unit  density  Treatment is  u n t i l t h e second  o f T04.  Treatment  thinned  be s c h e d u l e d  the  the harvest c u t ,  may be s c h e d u l e d  medium  Consequently,  period,  i s c u r r e n t l y 35 y e a r s  hectares  c o s t s and r e t u r n s t o  inaccessible.  (T04) i s t i m b e r  a t high  of  options,  c u t cannot  unit  hectares  are  (TG3), c o n s i s t i n g o f 10,000 h e c t a r e s , i s  management  harvest  Treatment  but  T08  harvest  unit  was p l a n t e d  T07  final  Treatment  decade o f t h e p l a n n i n g  T06  and  (10 $ / h a , ) .  by 3 $/cu.m. , and 2 $/cu.m., r e s p e c t i v e l y .  earliest  T05  factors  increased  T01,  T04  transportation  p l a n t i n g c o s t s t o 35 and 29 $/ha, r e s p e c t i v e l y ,  Similarly,  T03  However,  p l a n t i n g , and  unit  years  hectares.  8  (T08) i s i d e n t i c a l  older  t o T07 e x c e p t  (60 y e a r s ) . The a r e a  that i t  o f T08 i s 10,000  150  The  t i m e frame o f  the  planning  decrease  in  10 s t a g e s  e a c h have a d u r a t i o n  2 years,  early  those  network  management  three  regeneration  8695  feasible  management  unit  4,  opportunities contains In  only total,  represent  until  optimal the  t h e network  network policies  The  seguence found the  planninq  the  fourth  period.  actions  treatment  units  For  with  example,  20  years,  have  computes  formulation.  any  management  d e c a d e , and i t s n e t w o r k  management  model  seguences.  models f o r a l l  the  treatment  units  seguences.  Solution  by t h e SOLVE p r o g r a m s . The  i n F i g u r e 21 and F i g u r e  on e a c h t r e a t m e n t  the  period.  not  management  are represented  cf  have management  first  unit  represents  by t h e network p r o b l e m ,  projection  length.  (Appendix X) f o r T02, which has  does  Optimal  line  first  in  those  model was o p t i m i z e d  solid  The  to  10 s t a g e s a r e  sequences i n i t s network  28,341 a l t e r n a t i v e  volume p e r a c r e  period.  over  316 a l t e r n a t i v e  5.3.2 The U n c o n s t r a i n e d  Each  than  however, the  horizon.  a r e e a c h a decade  units that  report  options  defined  the next  i n the p l a n n i n g  outline  was  have many more a r c s and pathways i n  formulations  management  Treatment  period  actions later  of 1 year,  stages  treatment  i n the planning  their  the  r e s o l u t i o n towards the planning  and t h e r e m a i n i n g  Consequently,  problem  management  (The network model  the  over  optimal  and t h e sequence  assumes  the  22, a s  planning management  dashed  line  is  t o t h e end o f t h e that  the  optimal  T03: overmature, i n a c c e s i b l e second decade  1  1 2  1  1  1  1  1  1—™n  3 ^ 5 6 7 8 9  1  1  until  1  10 11 12  1  TO4: age 20, d e n s i t y  T  I  I  1 2 3 4  Decades  21.  Unconstrained optimal  I  I  5  I  ^  6 7 8  Decades  management sequences f o r treatment u n i t s  1-4  i  curve 9  I  I  I  I  9 1 0 11 12  I  41 T06: age 50 y e a r s , medium d e n s i t y (curve 89  T05: age 35 y e a r s , h i g h d e n s i t y (curve 9)  / i  /  "T  1  2 3  4  5  I  1  6 7 8  n  i ,  /  i / '  1  / i  1  "T  1  1  /  ~l  *•  "I  r~  "|  1  3 /, 5  6  r  7  8  1  1 2  9 10 11 12  1  ,  i  910  12  CD  i_  U TO7: age 50 y e a r s , curve 8, h i g h h a r v e s t c o s t s (+ $2)  3  24  [A 1  3  21  ~i  2  3  i  i  r  U  5 6  i  1  1  f  7  8  9 10 11 12  1  1  T08: age 60 y e a r s , curve 8, h i g h h a r v e s t c o s t s (+ $2) A  1  1  ~1  1  Decades F i g u r e 22.  Unconstrained  o p t i m a l management sequences f o r treatment  2  I  1  T  1  3  h  5  6 7 8  1  Decades u n i t s 5-8.  j-  9 10 11 12  r  153  management  Treatment  T01  seguence w i l l  timber  unit  cuts  harvest  Treatment higher  2,  is  at  cut during  first  at  a  decade l a t e r . Treatment  unit  high  standing  planning  4  (  >90  and 65 y e a r s ,  a  volume and  with  to  to a  high  cu.m./ha.) the f i n a l  7. 3 i s accessible timber  density.  A  with  cut occurs  ( a t t h e end i s c u t . The  medium  of unit  intensity  a heavy t h i n n i n g a  a t 85  grows u n d i s t u r b e d  period.  by  scheduled  and r e g e n e r a t e  a t 55 y e a r s  The h a r v e s t  unit  followed  similarly  timber  decade  i s scheduled  density.  (60 < cu.m./ha. < 90)  greater is  45 y e a r s  the  high  and 5 5 y e a r s ,  decade), the standing  planted  thinning  the  the  However, two heavy t h i n n i n g s  As soon a s t r e a t m e n t the  intensity  costs,  i t s standing  scheduled  harvest  to  i t s standing  years.  with  harvesting  density. are  c u t a t 65  to l i q u i d a t e  plant  o f medium  unit  liguidate  T04  and  t o be made a t 45 y e a r s  final  T03  1 i s scheduled  immediately  Thinning  TO2  infinitely) .  Unit  Treatment  are  be r e p e a t e d  until  years. 30 y e a r s  A medium t h i n n i n g , a l i g h t  into  thinning  ( <60 cu,m./ha.), and a medium t h i n n i n g a r e p r o j e c t e d f o r 30  years,  clearcut T05  38  years  occurs  a t 55  Treatment  unit  15 y e a r s ,  followed  final  harvest  and  45  years,  r e s p e c t i v e l y . The  years,  5 i s t o be t h i n n e d  (medium  intensity)  by a heavy t h i n n i n g a t 21 y e a r s ,  c u t a t 35  years.  at  and a  154  T06  Treatment thinning The  T07  T08  unit  final  harvest  optimal  that  o f T06, e x c e p t  year  25.  An  by a heavy t h i n n i n g  8,  volume  thinning  with  scheduled 23a.  period  liguidated.  when  These  unacceptable. unconstrained  the  i n the current  the  p l a n , $89.8 x10&,  c o n s t r a i n e d volume  until  scheduled  c u t a t 15  to  for  years.  f o r h a r v e s t i n g i n e a c h decade i s  would  first  2  net an  i n the  decades timber  probably  provides  flews.  occur  standing  present  one can compute t h e o p p o r t u n i t y  involving  years.  i s identical  is  Large f l u c t u a t i o n s  fluctuations  However,  9  years.  cut  a harvest  o f f the u n i t , e s p e c i a l l y  planning  a t 21  at  light  that the harvest cut i s delayed  heavy  unit  i n Figure  f o r an i m m e d i a t e  management s e g u e n c e f o r T07  immediate  presented  scheduled  cut occurs  The  The t o t a l  which  is  cut, followed  treatment  flow  6  make  of  the  i s being the  worth upper  volume  of  plan the  limit  with  c o s t s o f management  plans  155  1412-  unconstrained optima for each  Q  \ treatment unit  108-  \  <> a  6421—i  1  1  1  1  1—i  1—i—  1  3  4  5  6  7 8  9 10  2  o  first feasible s o l u t i o n  x  K H  _ scaled marginal values of volume per acre  8 u  (b)  6 42  a> +-  1  l/t  2  3 4 5  6 7 8 9  10  >  o X  E  final,"optimal 10  O  >  solution 40 [-36  8H 6 4  32  'O—°I  /  +  h28  CO  2  24  \  —i  1  1  1  1  1  1  1  1  2  3 4 5 6 7 8 9  1  r  0) C  (c)  £<o  «°  a> «/»• Z —  20  10  Decades Figure  23.  Volume plans.  flow  graphs  for  three  management  u n i t  156  5.3.3  Start-up  A linear Timber to  RAM  volume cut.  matrix  the present  cut  each  The c u r r e n t  the  RAM  corresponding  a  was c r e a t e d  was  a planning imposed  decade c u t was s e t a t 5 x 1 0  LP  were  were  omitted.  optimal  constraints  vectors  but c o n t r i b u t e d  that  large  the  decade  generated;  The LP v e c t o r s  management  i n MPSX f o r m a t  o f 10  cu.m./ha.  6  elements o f the matrix  activities  artificial  the  limit  d e c a d e t o w i t h i n + 10% o f t h e p r e v i o u s  t o the unconstrained  of  period  to  sequence  by t h e network  These v e c t o r s were s u p p l i e d t o t h e m a t r i x  set  with  {Navon, 1971b), The LP o b j e c t i v e was  n e t worth o v e r  a u t o m a t i c a l l y produced  system.  Feasible Solution  constraint  the s t r u c t u r a l  Timber  were  generator  A volume f l o w  Only  And F i r s t  model o f t h e management u n i t  maximize  decades.  Procedure  together  satisfied  negative  SOLVE  the  penalties  with  problem to the  objective function. The  MPSX  decomposition  program  been  smooth The  the is  decompositions,  eliminated  from  the  marginal  values  on t h e g r a p h  of  the  value.  a relatively  T02  that  Figure  ofthe  variables  23b shows t h e  the planning  produced  each  by a r r o w s ; t h e d i r e c t i o n  arrow r e p r e s e n t i n g , r e s p e c t i v e l y , marginal  iterations  artificial  solution.  volume  period.  decade  large  penalty  of  t h e s i g n and magnitude o f  against  volume  volumes o f s t a n d i n g  must be l i q u i d a t e d  are  and l e n g t h  Hence, from t h e g r a p h one c a n s e e t h a t  d e c a d e 1, due t o t h e e x c e s s i v e and  all  t r e n d i n volume c u t p e r decade o v e r  represented the  f o rthree  procedure.  After three had  was e x e c u t e d  before  year  there  production timber  in  o n T01  20 o f t h e p l a n n i n g  157  period.  Positive  a shortage The  value  t h e shape o f  decomposition,  the  Final  flow  displayed  system  was  both  values  restarted of  curve.  will  nine  the  next  change  i n F i g u r e 23c. Note t h a t  The  the  six  more  iterations.  o f f t h e management u n i t a r e  net revenue,  with a flow  decomposition  t h e network s u b p r o b l e m s  for  decomposition  weighted  policy  as a  commodity,  analogous  algorithms  to the  would  by t h e m a r g i n a l  then  values of  volume and n e t r e v e n u e . nine  the ninth  decompositions, decomposition  t h e problem  had p r o d u c e d  which had been i n c o r p o r a t e d i n t o  decision  to  examining  a plot  iterations. After  only  forces  t h e network  that  executed  seguences  LP  as  At  weight  seguences  and  o f volume and n e t r e v e n u e  have been c c n s t r a i n e d  After i.e.,  flow  regarded  manner.  for a total  volume c o n s t r a i n t s . solve  volume  be  Solution  decompositions,  could  can  p r o c e s s t o f a v o r management  The  The  arrows  the  indicate  decades.  marginal  curve i n the i n d i c a t e d  5,3,4  v a l u e s a t d e c a d e s 3, € and 7  o f volume i n t h e s e marginal  modifying  solution  marginal  stop  nine  profitable the  decompositions  o f t h e LP o b j e c t i v e  was  function  LP  management basis.  made  The  through  value against  the  (Figure 24).  seven  slightly.  at  had n e t c o n v e r g e d ,  decompositions,  Although  the objective  i t i s possible  that  value i n c r e a s e s  the procedure  simply  158  Iterations Figure  24.  S t o p p i n g r u l e - a s y m t o t i c b e h a v i o r o f the model o b j e c t i v e f u n c t i o n .  decomposition  159  found  a p l a t e a u on  further  substantial  model i s v e r y optimal  objective  cf  upper bound  it  optimal  $89.fi  optimal  surface  improvement,  close to i t s  return  constrained its  the  x10  is  policy.  return. Proximity  The  an  of the  increases confidence  could  undergo  more l i k e l y  provides  6  and  that  the  unconstrained  upper  bound on  the  o b j e c t i v e value  to  i n this  heuristic  stopping  rule. , On  examining  treatment  units,  10,000 h e c t a r e s flow  constraint  these  treatment  As  they  policies  i t was  found  selected that  322  of T03  are  left  of ±  10%  fluctuation  u n i t s cannot  a r e now  standinq period,  the  defined,  timber  within  when T01  has  liquidate  their  first  two  opportunity 74,2  =  c o s t of the  $15.6  exclusion  x10*) ,  of areas  attributed  to  constraints  with  excluded  areas  + 9.6)=) $6 The  selected  x10*  timber  first  cn t h e s e  few  optimal  of  volume.  units to  cutting,  be  A  held  or t o r e l a x t h e  decades.  Note  that  the  Although  most of t h e  and  T03, of  significant satisfying  with  return  as t h e  for  c o s t i s due  ($96  their x10 )  the  Even  individual  -  to  the  can  be  volume  would l e a v e  6  opportunity  cost  the  management s e g u e n c e s .  managed the  a  management s e q u e n c e s f o r e a c h as  surfeit  planning  c o n s t r a i n t i s s i z e a b l e (89.8  inefficient  management s e g u e n c e s , (74.2  the  necessity  were  d e c a d e s of t h e  even f l o w  o f T02  the  must  produced a  or f o u r t h decade b e f o r e the  decade,  T03  third  in  even  and  until  policy  the  unit.  be  flew  and a l l  management  might  even  h e c t a r e s o f T02  i n volume c u t p e r  solution the  to allow  individual  i n c l u d e d i n the  T02  already  the  unmanaged. B e c a u s e o f  be  the  for  flow  i f  the  optimum (89.8  cost.  treatment  manaqement  unit  unit  t h a t were  plan,  are  160  represented alternative with  terms  of  management  the areas  Treatment  T01  in  volume  sequences  that they  on  Treatment  unit  thinning resulting  is and  1  has  two  by  subseguent  unit  a  are  labelled  volume  2  version. thinning unchanged  Both  alternative  management  thinning  and t h e t h r e e  medium  alternative  to  Treatment  unit  seguence.  It  seguence  in  5  light  decade,  management  unconstrained  delay  the  optimum, a t 75  most  at  One  is  the  set  the  delayed  t h i n n i n g s have  seguences d e l a y s  55  years.  30 y e a r s  has been  been of  first  cut i s  complicated  occurred  thinnings.  differs both  the  seguence,  heavy  from  one  seguences. In g e n e r a l ,  management  t o 65 y e a r s  delays the  alternative  sequences  which  optimum  (13,250  but t h e s u b s e q u e n t h a r v e s t  has  activity,  unconstrained years,  4  by  i s the optimal  from t h e u n c o n s t r a i n e d unit  o f T01  that  cuts  two  management  Treatment  Most  management  yields.  has  by one decade  management  optimal  seguence  harvest  sequences, n e i t h e r o f which  T05  and 26. The  alternative  unconstrained  managed  i n greater  Treatment  cut  figures  i s a p p l i e d t o 1750 h e c t a r e s .  hectares)  two  the  25  Unit  seguence  T04  Figures  manage.  s e g u e n c e s . The o r i g i n a l  T02  in  of  first i n the  until  35  reduced t o the  the f i n a l  three harvest  years. managed from  by  the  timing  a  simple  management  unconstrained  and i n t e n s i t y  optimal  of thinning  I* TOl  3 ,i 13270  2 to  r  ZD U  ~i  i  i  i  1  J—^—i—i  1  2  3  U  5  6 7  1  8 9  1  1  1—  "i  1  10 11 12  I  i  i  I  I  2  3  U  5  6  1  I — i  1  r  7  8 9 10 11 12  CD  u  CD  / 4  TO 4  °  3  CD  E  2  /  A  // /  / | 11000  /  /  J ]/  /  /  |  .  15000  1  1 5250 3750-* 1  1  2  3 U  5  6  7 8  D e c a d es F i g u r e 25.  9 10 11 12  i  i  i  (  i  i  i  i  1  2  3  U  5  6  7  8 9 10 11 12  Decades  C o n s t r a i n e d o p t i m a l management sequences f o r treatment u n i t s 1,2,4, and 5  1  1  1  1—  4 TO 7 5750 CN1 O  Z  3  TO 6  2  4250  2  7950  H -i  1  i—i  2  3  1  1  5  U  1  6  1  7  1  8  1  9  1  1  1—  10 11 12  2050  i  i  r-  1  2  3  1  i  i  i  k  5  6 7  Decades  0; CD  o>  3  £  2  T08 4900  H  D  I  O  >  1  5100  1  1  i  i  2  3 U  i  i  5  1  1  1  6 7 8  1  1  1  1  9 10 11 12  D e c a d es Figure  26.  C o n s t r a i n e d o p t i m a l management sequences f o r treatment  u n i t s 6, 7 and 8.  1  1  1  1  1—  8 9 10 11 12  163  c u t s as w e l l as d e l a y i n g the f i n a l  harvest  cut  by  one  decade. T06  Treatment seguences. that  of  the  delay  has  two  the f i r s t  alternative  sequences, the t h i n n i n g cuts  and f i n a l  clearly two  alternative  management  the i n i t i a l  cuts.  The s e q u e n c e s d i f f e r  final  As w i t h for  I n both  past  management  harvest  cut  are  o f T07  both  spaced.  delay  the  management  t h i n n i n g one d e c a d e  optimal.  The  T08  Eoth  6  unconstrained  quite T07  unit  thinninq,  sequences  allowinq  heavier  thinninq  by one d e c a d e i n t h e t i m i n q o f  harvest cut.  T06 and T07, t h e a l t e r n a t i v e  treatment  unit  d e c a d e , and p r o v i d e  8 delay  manaqement  the i n i t i a l  the opportunity  to  sequences  t h i n n i n q by one  delay  the  final  harvest c u t .  In that  general,  are optimal  management optimal  unit  the within differ  sets of a l t e r n a t i v e the  manaqement  substantially  management  context from  management s e g u e n c e s f o r e a c h t r e a t m e n t  of  the  sequences the  whole  unconstrained  unit.  164  5,4 A G o a l P a r a m e t r i c  As s t a t e d is  usually  Analysis  i n Section  insufficient  With The D e c o m p o s i t i o n  4,1,3, knowledge  information  Model  of the o p t i m a l  f o r the decision  policy  maker. Some  knowledge o f t h e b e h a v i o r o f t h e s y s t e m s i n t h e n e i g h b o r h o o d the  optimum  described  above,  necessary cu.m.  is  to  usually one  ensure  to  the  technigue described more  insight  policy  Goal  the  to  the  programming  response  to  criterion,  such  With g o a l  the  c u t does n o t f a l l be  to  parametric  trade-offs  below 5 x 1 0  simply  model and r e - o p t i m i z e .  add  6  this  However, t h e  analysis,  provides  d y n a m i c s c f p r o b l e m s , by e x p l o r i n g t h e  the  present  optimum  the  terms  was i n t r o d u c e d difficulty of  and  the  objective  simultaneously, d e c a d a l volume  to the forestry  cf  maximizing  as p r e s e n t n e t worth  expressing cr  newly  1973)  function  literature management  minimizing  (Field,  programming, t h e o b j e c t i v e  of various c r i t e r i a .  A  explore  optimum.  in  The  wish  below, g o a l  between  objectives  goal  that  linear  into  space  ccnstrained  sum  might  F o r example, i n t h e p r o b l e m  e a c h d e c a d e . One a p p r o a c h might  constraint  in  reguired.  of  1 4  a  single  .  i s a weighted  F o r t h e p r o b l e m a t hand, an i n t e r e s t i n g  would be t o maximize t h e n e t p r e s e n t minimize t h e weighted n e g a t i v e  worth  and,  deviations  of the  be made t o t h e l i n e a r  model.  f r o m t h e g o a l o f 5 x10* cu.m.  few s t r u c t u r a l  c h a n g e s must  minimum c u t c o n s t r a i n t  described  above i s  See L e e (1972) f o r a pragmatic introduction to goal programming. Bell (1975) and D r e s s (1975) p r e s e n t e d t y p i c a l a p p l i c a t i o n s o f g o a l programming i n f o r e s t l a n d s management. 1  4  165  C. >  5 xlO  Formulating explicitly the  this  constraint  d e f i n i n g the  commodity  xio  incorporating  U  i n t o the  K  x ,  u k  W  is  goal Eg.  a  parameter  in  the  (5.4-3)  m o d e l , Eg. The  that  the  goal  8R  is  the  accomplished  negative  by  deviation  of  value,  1,  2,  objective  J  (5.4-2)  function.  J  u  -WED. j  uk  }  governs the function  objective  (5.4-3)  3  1 5  .  relative Note  function of  importance of that  the  the  when  H=0,  original  linear  (5.1-5) . procedure  increase  the  positive  real  followed  parameter  for  8 over the  number above which  solution.  The  used  maintain  to  ^  objective is  a  j =  Maximize { I E  (5.4-1)  2,  v a r i a b l e D.. as  6  D.  1,  as  from i t s d e s i r e d  c. + D. ^ 5  and  j =  parametric an  no  programming  optimal  this  demonstration  was  to  range [ 0 , P ] ,  where P was  the  change c o u l d  occur  facilities  s o l u t i o n . At  of  i n the HPSX  IP  were  f i x e d i n t e r v a l s of  W  True goal programming allows for ordinal solutions, i . e . goals are ranked r a t h e r than weighted, and each goal is optimized in o r d e r . The f o r m u l a t i o n u s e d i n t h i s s t u d y p e r m i t s trade-offs between the two goals of maximizing PN H and m i n i m i z i n g t h e s h o r t f a l l from t h e d e c a d a l volume g o a l . l s  166  (W=.5) t h e p r o c e d u r e  was i n t e r r u p t e d  performed  new a d v a n t a g e o u s management s e g u e n c e s . When  the  t o supply  parameter  ascertained for  reached  that  a  of  i s that  ten at less  decades unit  five  to the  per cent  The problems  Dantzig-Wclfe  o f the goal  when  W=.5  .  i n decades PNW t h a n  The nine in  o f t h e management  x10*, so  volume g o a l  c f t h e area  the cost  was $2.1  of  x10 . 6  o f t h e management u n i t was constructed  parametric  during  analysis.  A conventional  model l i n k s  t h e most e f f i c i e n t  o f commodities  formulation  Dantzig-Wclfe  n e t worth  $72.1  decomposition  and u t i l i z e s  t h e optimum  network  was  cccur  And D i s c u s s i o n  allocation  while  t o e i g h t , when  t o t h e management s e g u e n c e s  s i n g l e decomposition  5.5 Summary  the  policy  t h e minimum d e c a d a l  be s u b j e c t e d  system  satisfied  to t h e o b j e c t i v e o f maximizing  t o e i g h t . The p r e s e n t  Twenty-six  MPSX  was  on F i g u r e 27.  t h e volume c a n be i n c r e a s e d  cost  under t h e new  attaining  the  i s completely  W=,97, b u t u n s a t i s f i e d i n d e c a d e s f i v e  and  decomposition  W=.97,  a r e summarized  g r a p h shows t h a t t h e g o a l  implication  a  no f u r t h e r change i n t h e s o l u t i o n would  any l a r g e r w, R e s u l t s The  value  and  managment that  formulations  (MP1) i s s o l v e d seguence  t h e MP 1 and MP2  a s an  linear  decomposition  programming  LP  (MP2) i s f o u n d  e x p l o i t s the multistage problem  model d e s c r i b e d  nature  problem  through a o f MP2.  eguivalent  above  28,341 a l t e r n a t i v e management s e g u e n c e v e c t o r s ,  f o r each;  would  c f the contain  A measure o f t h e  Objective:  maximize  PNW + w(-Zdv )  o A  w= 0 w =..S  Decades  F i g u r e 27.  Decadal volume h a r v e s t computed by g o a l p a r a m e t r i c programmi  168  efficiency vectors  of the decomposition  i s the area of  vectors  resulting  histograms by  from  management  the  most  unit  recent  created  apparent  from  vectors  produced  utility  at  the  F i g u r e 28 t h a t  indicated  most o f  measure  of  unit  the  is  The  managed It  managed  is by  indicating  efficiency  model c a n be made t h r o u g h e x a m i n i n g  computer c o s t s f o r t h e d e m o n s t r a t i o n  problem  T a b l e 6. Computer c o s t s f o r a d e m o n s t r a t i o n Dantzig-Hclfe decomposition. Cost Component  UBC B a t e c o s t ($)  Commercial cost ( $ )  26.90 51.00 70.00 82.50  67.00 127.00 175.00 206.00  V i r t u a l memory Processing SOLVE s y s t e m Total cost  * Commercial rates times the university  Of t h e t o t a l  seguences  unit  decomposition.  the  the  decomposition.  a t t h e most r e c e n t d e c o m p o s i t i o n s ,  relevant  decomposition  SOLVE  to  of the process.  fi more  the  assigned  i n F i g u r e 28 r e p r e s e n t t h e a r e a o f t h e  vectors  the  the  p r o c e s s i n g e n e r a t i n g u s e f u l IP  f o r each  (Table 6 ) .  problem  of  Percentage of t o t a l cost 32% 62% -85%  of optimization,  851 i s  which  the  treatment  computes unit,  the  a breakdown o f  on 0BC«s IBM 370-168 a r e a p p r o x i m a t e l y rate.  cost  subsystem  1  of  at each  attributable  optimal  2.5  to  management  decomposition.  This  first feasible solution,  3_|  3 d e c o m p o s i t ions  0  1  2 3 4 5 6 7 8 9  10  2 3 4  5 6 7 8 9  10  1 2 3 4 5 6 7 8 9  10  0 1 2 3 4 5 6 7 8 9  10  3 7 decompositions 2  o  0 1 3  final solution, after  9  to cu  deco m p o s i t ions  2  rd  (_>  a> I i — i  0 2i  goal p a r a m e t r i c s , 1 decomposition (#10 )  Decomposition Figure  28.  Area  (%)  sequences  of  management  c r e a t e d  a t  u n i t  each  Iterations  s u b j e c t  to  the  decomposition.  management  170  series  o f programs i s h i g h l y  subroutines  of the three  stand  study.  The SOLVE s y s t e m  entry  p o i n t s and i n p u t - o u t p u t  as The  a stand high  be  with  intermediate  Heprogramming  f o r operational  Even system solution  with  unlikely $82.50.  very  that  virtual  in  use c o u l d this  as  be c r e a t e d the  vectors. and l o a d e d  the  due t o t h e  i t t o be used problem.  total  costs)  memory  management i n  the  could  solution  process.  s u b s t a n t i a l l y reduce the  part o f t h e system.  inadeguacies  favorably  in  t o s t o r e t h e network  with  described  the  - an LP model e n c o m p a s s i n g  seguences  could  of  are required  with  costs  o f t h e MPSX m a s t e r  results  the recognized  approach  management vectors  associated  compares  utilized  that enable  (32%  improved  p r o b l e m s and  costs  models  options  memory component  reduced  incorporating a l l the  i n c u r s f u r t h e r overhead  SOLVE s y s t e m . L a r g e a r r a y s  processing  growth  system, independent  virtual  probably the  alone  generalized,  Even into a  costs of optimization  only  above, t h e alternative  a l l the a l t e r n a t i v e  i f  a matrix  computer,  o f 28341 i t  seems  would be a s l i t t l e  as  171  6  *  Conclusions  Planning  is  organization, level  and  manager  planning  the  exerts  and  manager's  influence  is  often  models  conceptual  analysis  of  problem  that  S  formal  improves  technology  manager area  the  of  forest  biological Modern  and  This  thesis  analysis  i n s i g h t s from  components is  less  on  the  to  detailed  analysis,  fit t h e identified (land,  underlying  management u n i t as  the  allowable  unit  level  the are  the  decision  Examination  of  to  the  l i n e a r models ccmmonly and  to  models.  the  aid  the  In  the  simulate  two  problem  the  problem. at  the  treatment  unit  l e v e l s of at each  the  the  level  objective  of  problem  was  decision  a l l o c a t i n g constrained  c a p i t a l , etc.)  acceptable  i n t o the  structure.  level,  problem o f  cut,  management a c t i o n s . showed t h a t  mathematical  fact,  Instead,  performed  with  the  model.  arisen  these  decision  is  decisions  decision  sometimes  The  model,  has  at  high  collected,  problem. In  a  the  often  any a  'model'  formal  of  concentrates  systems  are  models e x i s t to  process, subjecting  elucidating  the  decision  planning a  data  r e s u l t s from  planning,  and  that  process i s formalized,  the  analysis  in  organization.  manager's c o n c e p t u a l  these  level,  his  but  the  decision  economic  management u n i t level.  the  lands  decision  of  control  model p r o v i d e s i n s i g h t s  of  in obtaining  over  informal:  where t h e  r a r e l y made d i r e c t l y cn  of  through planning  abstracted, model  even i n o r g a n i z a t i o n s are  level  i t i s primarily  process  compressed  highest  alternative  commodities seguences  of  problem's c h a r a c t e r i s t i c s used  pragmatic  at  the  formulations.  management Decision  172  analysis  technology  utilizing At  linear the  i s roost c o m p l e t e  programming  treatment  management a c t i o n s  optimal  manner.  structure  into  general  problem  the  A direct  The  'gaming  developing analysed should  i n the  must  (in  fact, of  with  be  requires)  stand  The network  second  formulation,  constitutes generate  a feasible  the  models,  ideal  o f the  considered. with  simulation  cases  growth  models  user's  manager w i l l  optimum, r e d u c i n g  where  each  management  and  in  mind.  analytic  A d e s i r a b l e f e a t u r e of  the  embeds  for  Test  d e c i s i o n a n a l y s i s technigues  validity  approach  simulation  demonstrate t h a t stand  experienced  at or near the  highly  situations.  s e g u e n c e o f management a c t i o n s and an  search,  a  generally considered  desirable  solution  yield  seguences.  s e g u e n t i a l simplex  i s that i t starts  Presumably  and  The  Two  algorithm  it.  process.  growth  management  management  approach  between t h e  tractability  stand  multistage  optimal  analysis  into  constructed  Trade-offs  solution  an  the  environment  1  insight  with t h i s  be  a natural  in  framework o f a c o n v e r s a t i o n a l s u p e r v i s o r  allows  decision  a  has  period  i s to  model l e d t o a c o n s i d e r a t i o n o f  climbing algorithm, the  system  interactive  ).  d e c i s i o n problem  planning  e x p l o i t e d i n the  decision  primarily  evaluated.  implemented  providing  be  of computing  a p p r o a c h e s were  system.  the  the  level,  MAX-MILLION  of i n c o r p o r a t i n g c o m p l i c a t e d  models  was  RAM,  level, over  this  T h i s d e c i s i o n problem  that should  difficulties  (Timber  unit  schedule  at  the  best  guess  attempts to provide  search  seguence.  n e t w o r k a u t o m a t i c a l l y from  a  improve starting  model  through  the  Computer  a brief  of  effort.  simulation  pathway  a  this  in  a  network programs  outline  of  the  173  management separate  alternatives, step.  intervention,  and s o l v e  Unlike direct  and i s l e s s  the objective Planning  function  decisions  made i n d e p e n d e n t relationship efficiently  handled  a t t h e management  master  and r e q u i r e s  Existinq  operational,  such  m o d e l . The  linear  available,  state  as IBM MPSX.  attractive,  level,  solving  decision  are a p p l i c a b l e the  a  without  some  On  anomaly  optimum.  level  unit  cannot  level.  be The  the l i n e a r  no  but i s l i n k e d  to a s e t  unit  p r o c e s s . The  decision  into  the  automatically  from  combines  RAM,  programming  is  that  stand  three are highly  performed  models  decomposition  problems under  analysis  to forest  conversational  of  powerful developed  p r o v i d e t h e management  the a r t mathematical  the  of  intervention.  system  Forest  A  allocation  i s incorporated  p l a n n i n g systems  of  decomposition.  i t c a n be e x e c u t e d  as Timber  be c o n t e m p l a t e d The  Both  unit  far  by  provide  unit  commercially  programming  e s t i m a t e s o f r e s p o n s e s t c management  Furthermore,  could  as  decomposition  components.  detailed  handles  model t h e t r e a t m e n t  the  The  runs  the t r u e  Wolfe-Dantzig  unit  system  such  f a r from  f o r m u l a t i o n o f the subproblem  LP s y s t e m ,  c u t on  at the treatment  problem  decomposition  and  to stall  surface,  through  problem  network  in  t h e two l e v e l s o f t h e p l a n n i n g p r o c e s s i s  program  that  problem  t h e system  a t t h e management  linear  subproblems  climbing,  likely  cf decisions  between  network  conducive t o post o p t i m a l a n a l y s i s .  t h e o t h e r hand, i t i s l e s s of  the  systems,  accurate  and  actions.  approach greater  i s economicaaly complexity  than  c o n v e n t i o n a l LP f o r m u l a t i o n s . technigues described  lands  planning  in  in this  British  s u p e r v i s o r and network  thesis  Columbia.  programs s h o u l d  174  be  used  to evaluate i n t e n s i v e  valuable  B.C. t r e a t m e n t  Douglas-fir.  of  confidence,  units,  developed.  the  management  of  action  intensive  decomposition  management unit  unit  management  profit  strategy  whole u n i t ,  managed, and  from  Much  optimize  in  timber  the  covered  by  the  normally.  a  policy  units. in  before s w i t c h i n g to decomposition A  the  in this  The  commodities  the  more  application  management  forest.  feasible  scheme. model  A  for  intensively  would  then  be  i t s c u r r e n t optimum,  Straightforward  the  management  RAM  u n i t s t o be  from  given  a portion of  sets of  problem  a  of the intensively  and  neighborhood  expensive,  but  inexpensive  LP  o f t h e optimum, more  complete,  analysis  techniques  procedure.  final  described  Only  through  alternative  decomposition  The  mode,  on  the  RAM,  some  PSYU o r T F I  of  the treatment  decomposition  managed t r e a t m e n t  models  obtained,  activities,  be t o g e n e r a t e a T i m b e r  including  coastal  improved.  for a coastal  t h e more c o m p l e t e  through  would  more  be made w i t h  model h a s been  augmented by t h e network s u b p r o b l e m s f o r each  finds  site  necessary  can  p r o v i d e d by T i m b e r  available  restarted  high  the  harvest  c a n be a d e g u a t e l y  would  practical  system.  sequences  seguences  the  costs  be c o o r d i n a t e d w i t h o n g o i n g  management u n i t  the  as  b u t s y s t e m s t o e s t i m a t e v a l u e must be  program  could  such  for  The e c o n o m i c components a r e l a c k i n g :  When an a c c e p t a b l e t r e a t m e n t a  policies  The b i o l o g i c a l components o f t h e  are a l r e a d y w e l l estimates  management  thesis of  the  non-timber and  of  the d e c i s i o n  pertains  t o the problem  of  timber  and n o n - t i m b e r  rescures of the  resources  brought  into  integrating  are  typically  treated  as  the  planning  problem  as  175  constraints.  However, u n l e s s a c o n c e n s u s can  value  resource,  of  timber  the  harvesting a c t i v i t i e s  The these  decomposition  trade-offs.  represented submodels action  i t cannot  on  i n the would  the  LP  non-timber  on  non-timber  decadal  vclume h a r v e s t e d . ) i n the  the  resource  utilized  as  impact  resource,  The  be  resource  master problem  simulate  on  dollar  with  the  objective function.  could  non-timber  action  represented  a  The  formed  compete f o r a l l o c a t i o n  i n the  system  be  to  evaluate  commodities  constraints, of  a timber  would and  a  might  impact  non-timber  management  resource  o b j e c t i v e f u n c t i o n as a w e i g h t e d  the  management  (Conversely, have an  be  on  the  could  be  goal.  176  BIBLIOGRAPHY Adams, D.M. and A, R. Ek. 1576, D e r i v a t i o n s o f o p t i m a l management g u i d e s f o r i n d i v i d u a l s t a n d s . P r o c , Soc. Amer, F o r . 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P r e n t i c e  pp. Hall, Inc.  W i l l i a m s , D.H., J.C. M c P h a l e n , S.M. S m i t h , M.M.Yamada, G.G. Young. 1975. Computer A s s i s t e d R e s o u r c e P l a n n i n g : an o v e r v i e w o f t h e CARP p r o j e c t , Dnpub. Rep,, B.C. F o r . S e r v . 30 pp. W i l l i a m s , D.H. And M.M. Yamada. 1976. C l u s t e r a n a l y s i s f o r l a n d management m o d e l s . Can. J . F o r . Res. A c c e p t e d f o r publication.  185  184  APPENDIX A  Land  Summary  of  I  Symbols  and  Notation  Units the  management  a  type  a  treatment  u n i t ,  A  treatment  u n i t  There Time  unit  i s l a n d ,  are  U  i  (e.g.  a  PSYU).  E.  E u  <—  i s  E.  an  treatment  aggregation u n i t s  i n  of  type  islands.  E.  Units t  the  time  The  planning  u n i t s Management  ut ut  i n t e r v a l .  (t  =  horizon  1,  2,  is  ...,  divided  T),  not  into  T  d i s c r e t e  n e c e s s a r i l y  of  time  equal  length.  A c t i v i t y a  management  the a  set  of  a c t i o n  a l l  management  treatment  i n  time  f e a s i b l e  sequence  unit  u,  i n t e r v a l  management  or  set  scheduled  of  t  on  treatment  actions  on  management  over  the  u  during  actions  time  from  set  of  a l l  management  sequences  f e a s i b l e  on  u.  the  set  of  a l l  management  sequences  f e a s i b l e  on  E.  candidate  set  the  treatment  the  simulated  of  management  u n i t s  u C  sequences  to  be  u. t.  f o r  T.  the  a  u n i t  applied  to  E.  States  ut  'jut  j u  beginning the A  of  amount  of  commodity  cut  i n  the  t o t a l  management i n t e r v a l  the  commodity  may  be  second  state  C. ju J  =  j  time  input  j u s t  p r i o r  u - l  T  Z  £  1=1  t=l  of  treatment  c  produced  s p e c i f i c  decade)  commodity  commodity  state  unit  u  at  the  t.  i i t 3  1  and  consumed the  independent  s t a t e , to  or (e.g.  u  defined being  on  volume of  as  the the  scheduled.  u  during  of  a c t i o n t o t a l  t.  timber time  t,  185  C. iut  the t o t a l commodxty i n p u t . . , up to time i n t e r v a l t .  s t a t e when u has been s c h e d u l e d  u-1 t C. = E C.. + I c. . -jut . . j i juk i=l k=l 1  J  M  the s i m u l a t i o n model a p p l i c a b l e t o treatment u n i t u .  u  { v  u(t l)' u(t+l) C  }  =  M  +  Commodity  u  ( { V  ut' ut ' ut C  }  a  )  Constraints  a^  a f e a s i b l e management p l a n f o r E.  g  a c o n s t r a i n t on commodity j .  Wt> "° Objectives R u  t  and Returns r e t u r n on a p p l y i n g u n i t a t time t . R  R  ut  = R  a management sequence  (v ,c ,a ) u t u t ' u t ' ut v  management sequence  return.  T R  R  = R ( v , c , a ) u u u u u  management u n i t r e t u r n  =  ER . ^ ut i  function.  U R = R ( c , a) = E R U=l  u  t o a treatment  186  APPENDIX I I I t e r a t i v e Approximation of F i n a l  Optimization  Conditions  of the m u l t i s t a g e model i n v o l v e s the f o l l o w i n g problem:  At stage U of MP1,  f o r each i n p u t commodity s t a t e C ^ ,  management sequence a^,  f i n d an  optimal  such t h a t the output commodity s t a t e C  will  be  i n the f e a s i b l e i n t e r v a l {LB.,UB.}. 3  i.e. find a  ^ R^Cv^c^jay ) is maximized, and  y  T C. + ^ c._ jU jUt  = C.  TT  t  =  and  ,  TT  JU  1  for all j = 1, 2, To  3  LB.< C. < UB. J - JU 3  J constrained commodities.  s i m p l i f y the d i s c u s s i o n , we  express A2-1  w i l l consider  The a  A -2  +  corresponding and  U  A ,A X  o n l y the lower bound  as the i n e q u a l i t y  V V V j>t " ™ £ ° S  A2-1  TT  2  Lagrange problem i s to f i n d  , ...,A /.  such t h a t J  L(a*,A) = R ( v , c , a ) + £ \ . g . u  u  u  u  (a^  A2-3  i s a maxxmum.  To m o t i v a t e the a l g o r i t h m ,  c o n s i d e r the h y p o t h e t i c a l s t a t i o n a r y  p o i n t s computed without r e f e r e n c e p l o t t e d on F i g u r e  below.  to c o n s t r a i n t j (Eq. A2-2), t h a t  are  187  a PI  ..  o,o I  PI:  J  o  P3  P2  29.  X.  P5 -  Figure  o-  Searching  f o r  complementary  slackness,  g. >0, A. = 0 3  J  PI  i s  an  i n t e r i o r  slackness  optimum.  c o n d i t i o n  i s  The  Kuhn-Tucker  complementary  met:  g.X. = 0 3 3  P2:  gj,<0,X_. = 0  The  s o l u t i o n  problem  P2  i s  c o n s t r a i n t  c o n d i t i o n  i s  met:  i n f e a s i b l e j .  w i t h  However,  the  respect  to  the  complementary  p r i m a l slackness  188  P3:  g  .  <0,  A . > 0 3  3  The  s o l u t i o n  slackness  P4:  g^>0,  The  does  not  8.X.  <0  s o l u t i o n  0,  =  P5  i s  a  respect searched Everett i s  a  to  used  .  is  p r i m a l  not  occur:  X  l i s t e d  j .  If  an  (1963)  has  shown  the  but  complementary  The  below  complementary  guide  f e a s i b l e ,  optimum.  the  as  complementary  complementary  slackness  holds.  c o n s t r a i n t  such  and  >0  .  boundary  algorithm  monotonically  i n f e a s i b l e  occur:  u n t i l  techniques be  s t i l l  >0  A  c o n d i t i o n  The  P4  does  g  g j  i s  >0  A  slackness  P5:  P3  that  tests  i n t e r i o r  slackness  for  A . x  the  simplex  of  \ j  >  u  '  optimum, does  c o n d i t i o n  f i x e d ,  f u n c t i o n  sequential  an  optimum  non-decreasing  s e l e c t i o n  for  i  =  1,  of  A  .  describe  not  i n t e r i o r e x i s t ,  occurs,  ....  J ,  i  g . A . 3 3 # i ,  Consequently, i n  Section  ^ j  w i t h >  =  are  u  0.  * g.(a ) 3 U T T  search  4.1,  can  189  Algorithm  Step:  1.  set  X.  0  =  3 f i n d  i f  a  a  g.. ( a ^ )  _>  otherwise,  2.  Select  3.  Compute  i f  then  new A .  a  3  a ^ 3  g.(a^)  g_. ( a y )  i f  gj(a^)  a^  i s  L  <0  go  maximized  to  step  4  >0  ^ u' a  then  ^ j ^  i  A_.  s  m  a  x  i  m  :  i  -  z  e  d  i s  too  s m a l l ,  go  to  step  2  i s  too  l a r g e ,  go  to  step  2  continue  >0  otherwise,  4.  0  i s  continue  otherwise,  i f  A J  L( {j>  3  u  then  A_  continue  =  0  optimal  then  go  w i t h i n  to  step  4.  c o n s t r a i n t  j ,  190  APPENDIX I I I  D i s t i n g u i s h i n g P o i n t s on a S t o c h a s t i c  The  o b j e c t i v e o f the a l g o r i t h m  evaluated  Surface  i s t o d e c i d e whether t h e r e t u r n  a t t h e p o l i c y p o i n t x-^ i s s i g n i f i c a n t l y g r e a t e r  r e t u r n a t x^.  than t h e  The r e t u r n f u n c t i o n R i s s t o c h a s t i c , and i s computed  w i t h a random number e~N(0,d).  R  The  algorithm  repeats  lj  = R(x , e.) 1 J  the v a l u a t i o n o f the r e t u r n f u n c t i o n w i t h d i f f e r e n t  random numbers, i n e f f e c t sampling the s t o c h a s t i c f u n c t i o n .  Sampling i s  repeated u n t i l a s t a t i s t i c a l t e s t o f s i g n i f i c a n c e i s passed, or a p r e s e t maximum sample s i z e i s reached.  On  e n t e r i n g the a l g o r i t h m ,  X  l '  V  X  R  the f o l l o w i n g i n f o r m a t i o n must be a v a i l a b l e :  the p o l i c y p o i n t s .  2  2 "  the e s t i m a t e d r e t u r n s  the c u r r e n t  f o r the p o l i c y p o i n t s .  sample s i z e t h a t the r e t u r n s R a r e  based on. n  max  o  - the maximum sample s i z e t o be allowed f o r e i t h e r policy. the standard  e r r o r o f e s t i m a t e of the r e t u r n  function.  z  the t a b l e v a l u e  o f the standard  above which the one t a i l e d HO : R., = R  0  normal d i s t r i b u t i o n  t e s t of the h y p o t h e s i s  w i l l be r e j e c t e d .  191  Steps  1)  s e t TEST = .FALSE.  2)  compute z =  (R-L  a  )  2  + _L n  3)  R  "  l  n  2  test of z > Z yes:  s e t TEST = .TRUE. go to step 9  no:  4)  continue  f i n d the s m a l l e s t sample i . e . f i n d k such t h a t n^ = minimum (n^, TI )  5)  test  i f maximum sample s i z e i s reached i f IL > n K —  6)  max  yes:  go to step 9  no:  continue  s i m u l a t e p o l i c y x^ w i t h a new random number  e  +  ^ and compute a  new average r e t u r n  \ 7)  Vk  =  +  R (  V\+l  ) ) / n  k  + 1  add new sample t o counter  n  k  \  =  +  8)  go to step 2.  9)  i f TEST = S  1  f .TRUE.  END.  >  F  A  L  S  E  -  R ^  i  > s  n  R o  t  2  significantly different  from  R  2  192  APPENDIX IV Listing  Of SIMOPT S c a n n e r  LISTING OF FILE 1 2 2 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48  SIMOPT.SCAN  09:5i  P.M.  OCT. 05, 1976  ID=PSYU  CONVERSATIONAL SUPERVISOR FOR OPTIMIZATION OF SINGLE STAND MODELS C C C DOUG WILLIAMS C JULY 8 1974 C UBC C C ^CONTINUE WITH IOEOCOM RETURN C C LOCAL STORAGE EXTERNAL GOULDtRMEYER,RKILK REAL P ( 1 0 , 5 0 ) , S T E P f 1 0 ) , O S SPAR(5J,1J) .DESV AR(50.10) INTEGER N P R ( 2 5 ) i N V R t 2 5 ) , L I S T C ( l u 0 J / 4 * l , 7 * 2 , 4 * 3 , 8 * 4 , 3 * 5 , 1 0 * 6 , 5 * 7 , 4 * 8 , 3 * 9 , 3 * 1 0 , 4 * 1 1 / * ,1 VARPtlOO) .IPARPUOO) LOGICAL TABSW L0GICAL*1 COMLST(IOO),COMAND<10) C C CONSTANTS NWRITE = 0 IPLUS = 0 LUY = 8 LUST = 9 C C SET UP CCMMAND LIST CALL MQVECl60,'READDISPLA YEDI TSI MUi-AT EOPTIMIZESUBROUTINEWRITESTOPSETFIXFREE' * ,CCMLST) C C READ COMMAND 25. CONTINUE CALL FR5ADI'SCARDS*,* STRING:•, CuMA,^Q,10) C C INTERPRET COMMAND CALL FINDC(C0MAND,10,' ' , 1 , 1 , N 3 Y T E , 1 O U M , 3 5 , 4 5 ) N8YTE = NBYTE-1 35 CONTINUE < CALL F!NDST(C0MLST,60,C0MAND,N3YTE,1.IBYTE,45.45) ICQM = L I STC(I BYTE) GO TO (1000,2000,3000,4000,5000,6000, 7000,8000,9000,10000,11000) ,ICOM 45 CONTINUE WRITE(6,200) CCMAND GO TO 25 C C C COMMAND : READ 1000 CONTINUE  49 50 51 52 53 54 55 56 57 53 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 85 90 91 92 93 94 95 96 97 98  C C C C  C C  .  READ P O L I C Y L A B E L AND L O G I C A L UNIT CALL F R E A O l , ' S : • * P L A B E L , 4 , ' I i • . L U P Q L l ) FIND P O L I C Y L A B E L DO 1025 1=1,1C00 ILINE = 1*100*1000 FINDILUPOLI'ILINS) READ(LUP0L1,100,END=1029) ALABcL I F ( A L A 8 E L . E Q . P L A B E L ) GO TC 1 0 3 5 1 0 2 5 CONTINUE 1029 CONTINUE WRITS16.210) PLABEL GO TO 25 READ P O L I C Y BLOCK 1 0 3 5 CONTINUE BACKSPACE L U P 0 L 1 R E A D ! L U P 0 L 1 , 1 0 0 ) I T T L E ( J ) , J = l , 10) DO 1045 1=1,1000 READ{LUP0L1,110,END=1049) IPAR I P A R P ( I ) = IPAR BACKSPACE LUP0L1 R E A D l L U P 0 L 1 , 1 2 0 1 PAR I I P A R 1 , ( D 5 S P A R l 1 P A R , J ) , J = 1 , 1 0 1 1 0 4 5 CONTINUE STOP 1 0 4 9 CONTINUE NP AR = 1-1 DO 1055 1=1,1000 R E A D ( L U P O L 1 , 1 1 0 . E N D = 1 0 5 9 ) IVAR I V A R P ( I ) = IVAR BACKSPACE L U P C L 1 R E A D l L U P 0 L 1 , 1 3 0 1 VAR ( I V A R ) , 1 T Y P t ( I V A R ) , R N G ( I V A R ) , B L ( I V A R 1 , 1 BU( I V A R ) , ( D E S V A R U V A R , J l , J = l , l u ) 1055 CONTINUE STOP 1 0 5 9 CONTINUE NVAR = 1-1 W R I T E ( 6 , 2 2 0 ) PLABEL,NPAR,NVAR GO TO 25  C C C COMMAND : D I S P L A Y 2 0 0 0 CONTINUE WRITS(6,230) ( T T L E ( J 1 , J = 1 , 1 0 ) DO 2 0 2 5 I=1,NPAR IPAR = I P A R P ( I ) WRITE16.240) I PAR,PAR(I P A R ) , t D 6 S P A R U P A R , J ) , J = l , 1 0 ) 2 0 2 5 CONTINUE  5  99 100 101 10 2 103 104 105 106 107 108 109 110 111 112 113 114 115 .116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148  WRIT6ie,250) DO 2 0 3 5 1=1,NVAR IVAR = I V A R P ( I ) WRITE(6,2603 IVAR,VAR11VAR),I T Y P E i I V A R ) , R N G l I V A R ) , B L ( I V A R ) , 1 BU(IVAR),<DESVAR(IVAR,J) ,J=l,6i 2 0 3 5 CONTINUE GO TO 25 C C C COMMAND : E D I T 3 0 0 0 CONTINUE CALL M T S C M D t ' $ E D I T POLICY",123 GO TO 25 C C C COMMAND : SIMULATE 4 0 0 0 CONTINUE NP = 0 TABSW = . F A L S E . CALL FREADI-2, 6NDLINE',0) CALL F R E A D ( ' * ' , « I : ' , N R E P ) I F I N R E P -EQ. 0) TABSW = . T R U E . I F ( T A B S W ) NREP = 1 IMOD = P A R ( 6 ) GO TO 1 4 0 2 5 , 4 0 3 5 , 4 0 4 5 , 4 0 5 5 , 4 0 6 5 ) ,IMQJ 4025 CONTINUE ITER = 1 X = G0UL0(P,10,1) GO TO 4 0 7 5 4 0 3 5 CONTINUE X = R M E Y E R ( P , 1 0 »13 GO TO 4 0 7 5 4 0 4 5 CONTINUE X = RKILK(P,10,1) GO TO 4 0 7 5 4 0 5 5 CONTINUE CALL INITL CALL EVCIN REWIND 2 X = V C M O L K P f 10, 1) GO TO 4 0 7 5 4 0 6 5 CONTINUE CALL I N I T L CALL INITC CALL EVCIN REWIND 2 REWIND 4 X = VCMDL21P,10,1) ,  ,_, ifl 0  1  149 150 151 152 153  154 155 156 . 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 173 179 180 181 182 133  4075  GO  5045  5055  190  191 192  50S5  TO  5085  CONTINUE CALL SIMPLXCF.P.10,NN,STEP,0,NITER,I,.001,VCMDL1,£5085) GO  5075  TO  CONTINUE CALL S I M P L X I F , P , 1 0 , N N , S T E P , 0 , N I T E R , 1 . . 0 0 1 , R M E Y E R , £ 5 0 8 5 ) GO TO 5 0 8 5 CONTINUE CALL SIMPLX{F,P,10,NN,STEP,0,NITER,!..001,RKILK,£5085) GO  5065  193  194 195 196 197 198  X  C C C COMMAND : O P T I M I Z E 5000 CONTINUE TABS k = . F A L S E . NP = 0 C C READ I N NUMBER OF I T E R A T I O N S AND R t P E T I T I O N S CALL FREADI-2,* ENDLINE',0) CALL FREADf•*<,*2I:'.NITER,NREPJ C A L L F R E A D C - Z t ' S N Q L I N E ' , * STREAM' ) I F 1 N R S P .EQ. 0) NREP = 1 C C LOAD S I M P L X B U F F E R S WITH F R E E VARIABLES DO 5 0 2 5 I=1,NVAR IVAR = I V A R P ( I ) I F I I T Y P £ ( I V A R ) . N c . 1 ) GO TO 5 0 2 5 NP = NP+1 P(1,NP) = VAR(IVAR) S T E P 1 N P ) = RNG<IVARJ I P ( N P ) = IVAR 5025 CONTINUE WRITE(6,410) ( I P f J ) , J = 1,NP) C C HAND A P P R O P R I A T E MODEL TO SIMPLX NN = NP+1 IMOD = P A R ( 6 ) GO TO ( 5 0 3 5 , 5 0 4 5 t 5 0 5 5 , 5 0 6 5 , 5 0 7 5 ) , I M Q O 5 0 3 5 CONTINUE CALL SIMPLXCF,P,10,NN,STEP,0,NITER,i...001,GOULD,£5035)  184  185 186 187 188 189  CONTINUE X = -X WRITE16.400) GO TO 25  TO  5085  CONTINUE CALL S I M P L X ( F , P , 1 0 , N N , S T E P , 0 , N I T E R , 1 . . 0 0 1 , V C M D L 2 , £ 5 0 8 5 )  C COPY NEW VARIABLES 5085 CONTINUE  BACK  TO  POLICY  199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 213 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248  5095  00 5 0 5 5 1=1,NP J = IP( I ) VAR(J) = P ( l , l ) CONTINUE PROFIT = - F W R I T E ( 6 , 2 7 0 ) N I T E R , P R O F I T , I P 1 1 , 1 ) , 1 = 1,NP) GO TO 25  C C C COMMAND : SUBROUTINE 6000 CONTINUE C A L L SUBLNK GO TO 25 C C C COMMAND : WRITE 7000 CONTINUE C C READ L O G I C A L UNIT OF OUTPUT POL ICY CALL FREADI'*•,•I:•,LUP0L2) C C WRITE P O L I C Y TO LUPQL2 NWRITE = NWRITE + 1 NLINE = NWRITE*100C00 FIND(LUPCL2'NLINE) W R I T E ( L U P 0 L 2 , 1 0 0 ) { T T L E C J ) , J = l , i 0) DO 7 0 2 5 I=1,NPAR IPAR = I P A R P ( I ) WRIT=(LUP0L2,280) I P A R , P A R { I PAR) , ( D i S P A R t I P A R , J ) , J = l , 1 0 ) 7025 CONTINUE E N D F I L E LUP0L2 DO 7035 1=1,NVAR IVAR = I V A R P ( I ) W R I T E 1 L U P 0 L 2 . 2 9 0 ) I V A R , V A R { I V A R ) ,1TYPE< I V A R ) , R N G < I V A R ) , B L { I V A R ) , 1BU(IVAR),(DESVARCIVAR,J j,J=1,10) 7035 CONTINUE E N D F I L E LUPOL2 GO- TO 25 C C C COMMAND : STOP 8C00 CONTINUE STOP ' C COMMAND : S E T 9 0 0 0 CONTINUECALL FREADI«S:•,T,1,•I,R:*,1ND,VALNEW)  £ -j  249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 230 281 232 233 284 285 286 287 288 239 290 291  9015  CALL F I N D C l T . l . ' V ' . l t l . I F I N . I F O , £ 9 0 1 5 1 V A R ( I ND) = VALNEW GO TO 25 CONTINUE P A R ( I N D ) = VALNEW GO TO 25  C COMMAND : F I X 1 0 0 0 0 CONTINUE CALL FRSADI'f'.'Ir'tlND) I T Y P S I IND1 = 0 GO TO 25 C COMMAND : F R E E 11000 CONTINUE CALL FREAD1•*•,•I:•,IND) ITYPE(IND) = 1 GO TO 25 C C FORMATS 100 FORM A T ( 2 0 A 4 ) 110 F 0 R M A T I I 3 ) 120 F0RMAT(3X,F6.0,T33,10A4) 130 F 0 R M A T ( 3 X , F 6 . 0 , I 3 , 3 F 6 . 0 , T 3 3 , 1 0 A 4 ) 2 0 0 FORMAT (/,'***• , 1 0 A 1 , ' CANNOT 3i£ I N T E R P R E T E D ***•,/) 210 FORMAT!/, ' * * * ' ,A4,• P O L I C Y CANNOT oc FOUND ***',/) 220 FORMAT (/,'POLICY ' , A 4 , ' , ' , I 3 , « PARAMETERS , ' , I 3 , ' V A R I A B L E S ', 1 • ARE READ I N . ' , / ) 230 F O R M A T ! / / / , 1 0 A 4 , / , 4 0 ! « . ' ) , / / , ' "ARAMETER ' • , l ' O E S C R I P T I O N',/« # VALUE',/) 240 FORMAT(I4,F8.2,7X,10A4) 2 5 0 FORMAT!//,' VARIABLE',T23,'STEP LOWER UPPER',/,' # VALUE 1 'STATUS SIZE BOUND BOUND D t S C R I P T I ON',/I 260 FORMAT!13,F8.2,16,F9.2,2F7.1,4X.6A4) 2 7 0 FORMAT!/,'AFTER' , 1 3 , '• IT E RAT IONS , THE BEST RETURN IS ' . F 9 . 3, / , 5X, • FR 1EE V 4 R I A B L E S : ' , 8 F 7 - 2 , / ) 2 8 0 FORMAT ( 1 3 , F 6 . 0 . T 3 3 , 1 0 A 4 ) 290 F0RMAT(I3,F6-0,I3,3F6.0,T33,10A4) 4 0 0 FORMAT!/,• O B J E C T I V E VALUE I S ' , F 1 0 . 2 , / ) 4 1 0 FORMAT!/,T17,'V A R I A B L E S',/,' *,« # Z* ',518) C C END oo  199  Simulated  Managed  Stand  Yield  Tables -  Meyers  POLICY  #1  YIELDS  PER  ACRE  OF  SITE THINNING ENTIRE STAND AGE (YEARS)  STAND  BASAL AREA SQ.FT.  TREES NO.  BEFORE  AVERAGE D.B.H. IN.  AND  MANAGED,  ilMDiX  L£VcLS= AFT2*  'AVERAGE HEIGHT FT.  EVEN-AGED  70,  20-YEAR  INITIAL  -  STANDS CUTTING  TOTAL VOLUME •CU.FT.  MERCHANTA B L E VOLUME CU.FT.  SAWTIM3ER VOLUME BD.FT.  4. 3 6.0  25 27  li*0 6J0  300 300  0 0  40  286  83  7.3  36  1230  930  0  50 50  284 172  107 73  8.3 9.1  45 46  1960 14o0  1680 1320  1800 1800  60  171  97  10.2  52  2070  193 0  4800  70 70  171 104  115 80  11.1 1L.9  59 60  2340 20lQ  2660 1910  9200 7500  80  104  96  13.0  65  27U0  2550  10500  90 90  104 67  111 80  14.0 14. 8  70 71  34J0 24dO  3230 2360  14400 10900  100  67  92  15.9  75  30aO  2910  14400  110 110  67 21  104 40  16.9 18.6  79 80  3630 14i0  3470 1360  13000 7400  120  21  47  20.2  34  1740  1680  9700  130  21  54  21. 7  86  20a0  2010  12200  TOTAL  MERCH. C U . 8D.  FT. -  FT . T R Ees  -  INCLUSION  IN T O T A L  TREES  INCHES  IO.O  6.0  INCHES  D. B.H.  Y I EL D S — D.B.H.  AND  -  3 2 0 . CUBIC LARGER  AND LARG-R T J  80. . PERIODIC  119 57  FOR  PINE  CYCLE  THINNING  950 283  CUTS  PONDEROSA  8 0 . , SUBSEQUENT  30 30  MINIMUM  OF  TO  8-INCH  TREES NO.  BASAL AREA SQ.FT.  TOTAL VOLUME CU.FT.  CUTS  MERCHANTA B L E VOLUME CU.FT.  SAWTIMBER VOLUME BD.FT.  662  62  560  0  112  29  500  360  67  35  810  750  1700  37  31  920  870  3500  46  64  2220  2110  10600  7090  6100  28000  YIELDS  F E E T AND  INTERMEDIATE  1 5 0 0 . BOARD  0  0 •  FEET  4-INCH T O P . TOP.  N  o o  POLICY  #2  YIELDS  PER  ACRE  Oh  SITE THINNING ENTIRE STAND AGE ( Y E A ° S)  TREES NO.  30 30  950 2 74  40  STAND  BASAL AREA SQ.FT.  BEFORE  AVERAGE D.B.H. IN.  MANAGED,  INDEX  LEVJ!LS=  AMD AFTER  AVERAGE HEIGHT FT.  EVEN-AGED  70,  20-YEAR  INITIAL  STANDS CUTTING  76.,  TOTAL VOLJME CU.FT.  MERCHANTA B L E VOLUME CU.FT.'  SAWTIMBER VOLUME BD.FT.  11*0 600  280 280  0 0  272  79  7.3  36  1170  880  0  50  270  104  8.4  45  19iO  1650  1500  60  267  126  9.3  51  2640  2410  4500  70 70  263 154  146 100  10. 1 10.9  53 59  34d0 24u0  3230 2310  8900 7300  80  154  117  11.3  65  3230  3030  12400  90 90  154 103  133 99  12.6 13.3  69 70  40u0 30^0  3780 2860  15300 12100  100  103  115  14.3  74  3 73 0  3540  16100  110 110  103 32  130 50  15.2 17.0  78 80  445 0 17,30  4230 1700  20400 8700  120  32  60  18.5  83  22J0  2110  11500  130  32  69  19.9  36  26TO  2 540  14600  TOTAL  MtRCH. CU. BD.  INCLUSION  F T . - TREES  F T . - TREES  10.0  6.0  INCHES  IN T O T A L INCHES  YIELDS—  D.B.H.  D.B.H. AND  ANJ  LARGiR  -  3 2 0 . CUBIC LARGER TO  100. PERIODIC  25 27  FOR  PINE  CYCLE  THINNING  4.8 6.0  CUTS  PONDEROSA  SUBSEQUENT  119 54  MINIMUM  OF  TO  8-INCH  FEET  4-INCH TOP.  TREES NO.  BASAL AREA SQ.FT.  6 7 6  6 5  1 0 9  4 6  1500.  TOTAL VOLUME CU.FT.  CUTS  MERCHANTA B L E VOLUME CU.FT.  SAWTIMBER VOLUME BD.FT.  5 9 0  1020  1600  9 2 0  51  34  9 8 0  7 1  80  2 6 7 0  2 5 3 0  1 1 7 0 0  7900  6910  3 1 1 0 0  YIELDS  AND  INTERMEDIATE  BOARD  9 2  0  3 2 0 0  FEET  TOP.  to o  POLICY  YIELDS  #3  PER  ACRE  OF  SITE THINNING ENTIRE STAND AGE (YEARS)  TREES NC.  STAND  3EF0RS  AVERAGE D-3.H. IN.  BASAL AREA SQ.FT.  AND  AVERAGE HEIGHT FT.  MAN4 GEO,  INDcX  LEV£LS= AFTE*  EVEN-AGED  70,  20-YEAR  INITIAL  -  STANDS CUTTING  TOTAL VOLUME CU.FT.  MERCHANTABLE VOLUME CU.FT.  SAWTIMBER VOLUME BD.FT.  4. 3 5.7  25 26  1190 3.>0  320 320  0 0  40  432  109  6.8  35  1570  1060  0  50  424  134.  7.6  44  24i0  1900  0  60  412  155  3.3  50  3190  2730  3300  70 70  395 209  171 107  8. 9 9.7  57 58  3 990 25o0  3560 2360  6600 5000  80  203  127  10. 6  64  3420  3190  9800  90 90  208 133  147 108  11.4 12.2  69 69  4330 S2dO  4060 3030  16000 11700  100  133  124  13.1  73  397 0  3760  15800  110 110  133 40  140 54  13.9 15.7  77 79  4730 13/0  4480 1780  20100 8600  120  40  64  17.1  82  2320  2220  11500  130  40  74  13.4  85  2760  2 670  14600  TOTAL  MERCH. CU. BD.  INCLUSION  F T . - TREES  F T . - TREES  10.0  6.0  INCHES  IN T O T A L INCHES  YIELDS—  D.B.H. AND  D.B.H. AND  LARGER  -  320. LARGER TO  108. PERIODIC  119 77  FOR  PINE  CYCLE  THINNING  950 436  CUTS  PONDEROSA  1 1 5 . , SUBSEQUENT  30 30  MINIMUM  OF  CUBIC TO  8-INCH  FEET  4-INCH TOP.  TREES NO.  BASAL AREA SQ.FT.  1.  TOTAL VOLUME CU.FT.  CUTS  MERCHANTA B L E VOLUME CU.FT.  SAWTIMBER VOLUME BD.FT.  514  42  360  186  64  1430  1200  1600  75  39  1110  1030  4300  93  86  2860  2700  11500  8540  7600  32000  YIELDS  AND  INTERMEDIATE  BOARD  FEET  TOP. to O to  POLICY  #4  YIELDS  PER  ACRE  OF  SITE THINNING ENTIRE STAND AGE (YEARS)  TREES NO.  STAND  BEFORE  AVERAGE D.B.H. IN.  BASAL AREA SQ.FT.  AND  AVERAGE HE IGHT FT.  MANAGED,  iNDEX  LEVeLS= AFT E*  EVEN-AGED  70,  20-YEAR  INITIAL  -  STANDS CUTTING  TOTAL VOLUME CU.FT.  SAWTIMBER VOLUME BD.FT.  119 75  4. 8 5.7  25 26  1190 8i0  40  417  105  6.8  35  I5d0  1030  0  50  410  133  7. 7  44  2390  192 0  0  60  399  154  3.4  50  31/0  2730  3500  70 70  384 206  170 108  9.0 9.8  57 58  39/0 25dO  3570 2380  6900 5200  80  205  128  10.7  34T0  3220  10200  90 90  205 131  148 108  11.5 12.3  69 69  4350 32JQ  4080 3 040  16500 11900  100  131  124  13.2  73  39d0  3770  16000  110 110  131 40  140 54  14.0 15.7  77 79  4 7J0 1370  4490 1780  20200 8600  120  40  64  17.1  82  23^.0  2220  11500  130  40  74  18.4  85  2790  2680  14600  310 310  0 0  TOTAL  MERCH. CU. BD.  FOR  INCLUSICN  F T . - TREES  F T . - TREES  10.0  6.0  INCHES  IN T O T A L INCHES  YIELDS—  D.B.H. ANJ  D.B.H. AND  LARGER  -  3 2 J . CUBIC LARGER Tu  108. PERIODIC  MERCHANTA B L E VOLUME CU.FT.  .  PINE  CYCLE  THUNING  950 '421  CUTS  PONDEROSA  1 1 1 . , SUBSEQUENT  30 30  MINIMUM  OF  TO  8-INCH  FEET  4-INCH TOP.  TREES NO.  BASAL AREA SQ.FT.  TOTAL VOLUME CU.FT.  CUTS  MERCHANTA B L E VOLUME CU.FT.  SAWTIM3ER VOLUME BO.FT.  529  44  380  0  178  62  1390  1190  1700  74  40  1120  1040  4600  91  86  2860  2710  11600  8 540  7620  3250C  '  YIELDS  AND  INTERMEDIATE  1. BOARD  FEET  TOP.  o  204  APPENDIX Simulated  Stand  VI  Tables -  Goulding  __L_C.Y___1 \GE  IU 11. 0  CU  TARIF  VOL 7. > 213  >  20 2 1 . 7  ** 2 0  3 92  4 116  5 96  STAND T A 3 L S : S I T E 9 .0 7 3 6 20 4 44 28  0 0  28 4  0 24  8 20  20 24  I ** DBH CLASS 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 2 3 29 3 0 31 32 3 3  0 iO 40 16 24  33.5 33. 5  1066 4G93  413 2400  50 3 7 . 0 50 3 7. 0  1198 5181  775 3818  8 0  8 4  0 20  39.7 39. 7  1507 5546  1024 4356  0  4  7 7  0 4  0 13 4 12 16  7 5  70 4 2 . 1 70 4 2 . 1  1555 5461  1334 5129  0  8  0 3  0 11  0 3  7858  7580  0  0  40 40  60 60  90  46.1  12 12 12  4 12  8  0 8  0 8  0 4  0 0  8 5 8 15  8 il 4 25  0 3  4 4  0 4  4 4.  12  R0L.LC.Y_32 >  7. > 148  IU 11. 0  20  21.7  38 38  32.7 32.7  1166 3812  148 2024  48 48  36.4 36.4  1826 4334  53 3 9 . 2 58 3 9 . 2 68 4 1 . 7  2 0  ** 5 3 4 200 200 112  STAND T A B L E 8 7 6 16 36 56  : S ITS 150 10 11 12 9  DBH C L A S S 13 14 15 16 17 18  24 36  36 32  8 64  0 4 4 20 40 4 24 16 16 3 -2  1048 3273  20 0  8 20  16 4  20 44  1889 4395  1228 3637  0  15  3 1  0 15 3 27  5808  4936  1  3  32 0  24 24  0 16 12 0 8 8  19  20  21  22  16  4 0 0 12 8 12 12 12 0  12  4  5 3  0 0  0 0  4 4  4 0  4 0  0 8  8 4  0 0 8 12  1 7  4  0 27  3  0  0  4  0  0  0  8  8  8  4  to o  Cn  AGE  TARIF  CU >  VOL IU 7. > 11. 149 0  20  21.7  47 47  36.2 36.2  2183 5471  938 3580  57 39.0 57 3 9 . 0  2490 5172  1668 3841  67 41.5  6876  6004  2 0  3 192  ** 4 5 200 108 4  24  STAND T A B L E 6 7 3 56 36 16  : S i r g 150 * * DBH C L A S S 9 10 • 11 12 13 14 15 16 17 18  20 0  0 40  44 4  4 4 32 52 32 0  0  16  8 20  4 24 0 16  12  8  0 32  0 20  0 20 24 4 0 12 0 0 27  0  0 4 3 12  16  12 12  19  20  21  8 8  0 0  4 4  8 0 4 12  4 4 12  8  0  4  4  4  4  8  0  8  22  23  24  8  8  4  to  o  207  APPENDIX V I I A M u l t i p l e D e c i s i o n T h i n n i n g P r o b l e m As A Network ( K i l k k i ' s Scots Pine Model).  The  management  demonstration  LISTING  OF  FILE  problem  15 9 7 6 5 4 3 2 1 10 10 SEND  Line  input  described  41. 53. 53. 53. 61. 63. 65. 67. 76. 93.  JULY  2  1974  end-of-flie defined, used to  card.  calibrate  activity  The  for  the  KILKKI•S  AUG.  04,  first  below:  ID=PSYU  1976  AODdL  0 9 6 7 1 4 3 1 4 3 1 2 3 1 1 1 2 1 10 1 1 10 1 10 1 0  0 0 0 0 0 0 0 0 0 0 0  title.  time  Lines  10 10  2-4 d e f i n e t h e t i m e  4 defines  frame  5 10 2 1 1 2 1 10 10  stages  definition  frame  3-30 t o be o f 2 i s ended  with  Any number o f management o u t l i n e s may t h e n  delineated  to i d e n t i f y  A.M.  41. 71. 71. 71. 86. 86. 86. 86. 91.  1 i s an i d e n t i f y i n g  duration.  set  0.0  o f t h e g r a p h . F o r example, l i n e years  data  i n the text i s l i s t e d  11:13  DATA.GEN13  T H E S I S EXAMPLES 1 1. 1 2 51. 2 30 2. 3 SEND S C O T C H P I N E , AGE 3  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  outline  Problem  by e n d - o f - f i l e c a r d s .  the treatment  feasible  9  to  transitions.  be  6, 7 and 8 a r e  u n i t and any p a r a m e t e r s  t h e s i m u l a t i o n model. L i n e s  s t a t e s and t h e i r  Lines  an  19  necessary  define  The o r d e r  the  of the  208  data  e n t r i e s corresponds to the t a b l e  outline  report  The  included  runstream  requiring  only  (DATA. GEN 13)  i s assigned  graphs  written  unit  are  below and d e s c r i b e d  f o r the graph  two f i l e  e n t r i e s o f t h e management  generating  assignments.  to  i n the t e x t . program  is  simple,  The management o u t l i n e  t o be l o g i c a l  file  u n i t 4, and t h e g e n e r a t e d  the s e g u e n t i a l  file  GFILE13 cn  logical  3.  # $RUN GENERATE 3=GFILE13 4=DAT A.GEN 13 0 # EXECUTION TERMINATED  After produced logical and  generating with  second  on  can  be  p r o g r a m . A g a i n , GFILE13 i s a s s i g n e d  to  u n i t 3. A s i n g l e d a t a c a r d  controls  create  a  the graphs  the  the type o f r e p o r t . management  below.  # $RUN REFORT 3=GFILE13 3 SEND  # EXECUTION  TERMINATED  outline  GFILE 13,  identifies  Report o p t i o n report  and  reports  the treatment 3  was  arc l i s t  used  unit to  included  209  TREAT KENT UNIT # 1 : SCOTS PINE, AGE 50 E N T R Y STATE FIRST LAST  15 9 7 6 5 4 3  J I I l J I !  2 ! 1 I  10 |  1 11 53 53 53 61 63 65 67 76  1 41 71 71 71 86 86 86 86 91  THE CORRESPONDING AND 1972 ARCS ARC  DURATION COSTS MIN MAX I AREA VOL  I | j I | | | | | |  I  VOLUME SIM CUT J I OUTPUT STATES  I I I I I | I |  1 | | | \ | J j | f  I  l  9 F | rr | 7  j |  j  I  I  I  T | | T ! l T |  I | |  I  I  I  T I TI T j  5 10 2 1 2 1 10 2 1 10 2 1 10 1 10 10 10 4 4 3  NETWORK HAS 29886 FEASIBLE MANAGEMENT  6 3 3  SEQUENCES  LIST  F R O M STAGE STATE 1 2 2 2 2 2 2 2 2 2 2 2 2 3 4 5 6 3 4 5 6 2 2 2 3  15 9 9 9 9 9 9 9 9 9 9 9 9 6 6 6 6 7 7 7 7 9 9 9 C  A T STAGE STATE | f } I I 1 1 ! 1 1 1 ! 1 1 1 1 | 1 1 1 | 1 I 1 1  2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 7 7 7 7 7 7 7 7 8  9 5 6 7 5 6 7 5 6 7 5 6 7 4 4 4 4 4 4 4 4 5 6 7 3  SIMULATE?  1  ! ! I  1 1  T T T T T T T T T T T T T T T T T T T T T T T T T  C O S T S VOLUME  210 A d d i t i o n a l treatment terminated A  with  unit  r e p o r t s c a n be g e n e r a t e d  an end f i l e  runstream  or  execution  command.  f o r t h e network s o l u t i o n  text.  off  GFILE13. l o g i c a l  units  the  LP d e c o m p o s i t i o n  model d e s c r i b e d i n S e c t i o n 5. At t h e end o f  solution  process,  o p t i m a l management s e g u e n c e the  same  vectors.  The  parameter  program  graphs  3 and 9 a r e a s s i g n e d  file  contains  below. The  (described i n  the  the  S e c t i o n 4.2.1) i s l i s t e d  program  -POL  information »SOLO»  i s t o be r u n w i t h o u t  to f i l e s  contains  a  f o r each treatment  as  signifies  that  read  used i n  summary o f t h e  unit,  expressed  linking  are  a t f i l e -LP  Timber  EAK  the  LP  solution  t o t h e L P program.  # $BUN SOLVE 3=GFILE13 8=-P0L 9=-LP PAE=S0L0 1 1 2 0.02 SEND # EXECUTION TERMINATED  A single unit to  graph.  type to  card c o n t r o l s the s o l u t i o n  The f i r s t  be l o a d e d  function  data  field  MAI, ENw,  of r e p o r t s generated. be  used  in  demonstration (and worth  only) at  decisions  terminated The  the  field  etc.).  The t h i r d  present  net  worth  loaded  2%  graphs with  stage  interest were may  rate.  A  generated be  an e n d - f i l e  loaded  c o n t r o l s the  For the  G F I L E was t h e f i r s t  stage  the  present  net  r e t u r n r e p o r t and a  are  and  objective  calculation.  from  and  i n GFILE  i s the d i s c o u n t r a t e  one, The o b j e c t i v e was t o maximize a  field  treatment  unit  defines the  The f o u r t h f i e l d  problem, t h e graph  matrix  Additional  d e f i n e s the treatment  and s o l v e d . The s e c o n d  (maximize  o f each  included  solved,  or  below. execution  command.  return report l i s t s  various  values  of  interest  211  for  each s t a t e  stage  and s t a g e  i s also given.  state-stage the  combination.  Five  values  The o p t i m a l  are recorded  at  state each  at each feasible  combination. maximum s t a g e  values  return  (constrained  the  objective  the  volume p r o d u c e d  the  capital  the  w i t h t h e LP commodity  decision derivative),  stage  return  on e n t e r i n g  on e n t e r i n g  produced  maximum  calculated  stage  the s t a t e ,  the s t a t e ,  (or consumed) on e n t e r i n g return  (unconstrained  the s t a t e , decision  derivative). The  decisions  combination, have e n t e r e d back and  the state.  matrix  the optimal  records  s t a t e and s t a g e  f r o m . The d e c i s i o n s  optimal  for that  matrix can  management s e g u e n c e  each  be  stage-state  t h e system used  to  from a n y ( f e a s i b l e )  should trace stage  S T A G E . OPTIMAL  1  .4  5.  R E T U R N S a  2  1 .03 0.  0.83 0.52  1.05 0.6t>  0.85 0.54  1.08 0.69  0. 87 0.55  1.11 0.7i  0. 89 0.57  1.12 0.72  0.89 0.57  1.13 0.75  0.90 0.59  -lfl.  .11.  .12.  .13.  .14-  .15..  15  0.?. 7 0.27 1.25 0.78 0.2 7 27 27 1.29 0.81 0.27 0.27  0.27  0.27 0.27 0.81 0.54 0.27 0.27  0.27 0.74 0.52 0.27  0.83 0.56 0.27  to to  1  0.27 1.44 1 .04 0.55  1 1  1 11  1 | |  5  j  1 12  13  1 | | | 1  5  1 | |  1  j  1 1.4  1  1  0.30  0-80 0.57 0.27  0.55 0.28 1.49 1.09 0.55  0.30  0.27  0.30  0. 82 0.60 0.27  0.55 0 . 23 1.53 1. 14 0.55  0.30  0. 27  0.30  0-8 3 0.61 0.27  0.55 0. 28 1.53 1.19 0.5 5  0.30  0.27  0.30  0.35 0.64 0.27  0.55  0.30  0.27  0.55  0.30  0.36 0.65 0.27  0.55  55  0.48 0.20 1.21 0.93 0.48  0.87 0.67 0.27  0.55  0.48  0.27  0.55  0 . 4 8  0.87 0.67 0.27 •  0.55  0.48  0.27  0.55  0 . 4 8  0.38 0.69 0.27  0.55  0.48  0.2 7  0 . 4 3  0.39 0.70 0.27  | | I  15  |  1  1  j  | 0.  1 16  1 | |  1  j 1 17  I  |  1  | 1 18  1  1  | | 0.55  0. 86 0.61  1-14 0.77  0.91 0.62  1.13 0.77  0.90 0.62  1.13 0.79  0.90 0-63  0.27  0.27 0.87 0.62 0.27 0.27 0.88 0-64 0.27 0.27 0.83 0.65 0.27 0.27 0.90 0.67 0.27. 0.27  0.27 0-89  0.66 0 . 2 7  0.27 0 . 9 0 0 . 6 3 0 . 2 7  0.27 0 . 8 9  0.68 0 . 2 7  0.27  0.82 0.27 1.45 1.18 0. 82 0.83 0.28 1.56 1.29 0.83 0.84 0.29 . 1 . 6 6 1.39 0.84 0.85 ' 0 . 3 0 1.77 1.50 0.85 0.66 0-31  1.88  0 . 8 9  1-61  0 . 6 8 0 . 2 7  0.86  214  ( ^ O  ro  —i  CO CO CO  o > ro o < ! • o0 ^ ^ in o co rg o C O co ro ro o co O O N H O oorg N O  CO  r-  rj  rvj  CO  -1-  ro  <7> CO  ___1_I__S.__AT_I_ FROM { STATE, STAGE ) _IA._E.___1 1 0 0 2 0 0 3 4 0 0 5 0 0 0 0 6 0 0 7 8 0 0 9 0 0 10 5 9 5 10 11 12 5 11 13 5 12 14 1 13 15 1 14 1 15 16 17 1 16 1 17 18 19 . 1 18 20 1 19 1 20 21 22 0 0 23 0 0  7 0 0 0 0 0 0 0 7 2 2 2 2 2 5 2 2 ?  2 2 2 0 0  0 0 0 0 0 0 0 3 9 10 11 12 13 14 15 16 17 18 19 20 0 0  S T A T E S.  _, 0 0 0 0 0 0 5 5 5 5 5 5 5 5 5 5 5 5 3 3 0 0  0 0 0 0 0 0 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 0  0 0 0 0 0 7 7 7 7 7 7 . 7 7 7 7 7 7 7 4 4 0 0  0 0 0 0 0 6 7 3 9 10 11 12 13 14 15 16 17 18 19 20 0 0  .11 0 0 9 2 9 3 5 4 5 5 5 6 •5 7 5 8 5 9 5 10 5 11 5 12 5 13 5 14 5 15 5 16 5 17 5 18 5 19 5 20 0 0 0 0  0 9 9 9 9 9 9 9 9 9 9 6 6 6 6 6 6 6 6 6 0 0  0 2 3 4 5 6 7 0 9 10 11 12 13 14 15 16 17 18 19 20 0 0  0 9 9 9 9 9 9 9 9 9 9 7 7 7 7 7 7 7 7 7 0 0  o• 2 3 4 5 6 7 9 10 il 12 13 14 15 16 17 18 19 20 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  15 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 0 0  0 1 0 2 3 0 4 0 0 5 6 0 7 0 8 0 9 0 10 0 11 0 12 0 13 1 14 1 15 1 16 1 17 1 13 1 19 1 20 1 0 1 0 , 10  0 0 0 0 0 0 0 0 0 0 0 0 13 14 15 16 17 18 19 20 21 22  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 c 0 0 0 0 0 0 0  12 0 0 0 0 0 0 0 0 0 0 0 G 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  1_ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ' 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0. 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  Ln  216  The 4.2.3 in  second  allows the  up t o t h r e e  previous  corresponding 7),  multiple  but  the  generally  as b e f o r e the  thinnings.  problem  entry  reduced  arcs  problem  were  Two  by  times 10  discussed  thinning  not  t o s t a t e 4, and DL = 3 . 0  graphs a r e about 29886  thinning  t h e same s i z e  considered  Consequently,  (see F i g u r e  management  F i g u r e  1 1 ) . The p r e c e d e n c e  outline report  30.  Precedence problem.  are included  graph  f o r  K i l k k i ' s  1.5 state  states  were  and s e c o n d  graph  =  to  the  and 29729 a r c s , r e s p e c t i v e l y . The s t a t e s  included  (DL  corresponding  f o r the f i r s t  section  levels  f o r t h e second t h i n n i n g years..  in  resulting problem,  are defined  (Figure 3 0 )  below.  demonstration  and  217  # E EEPOET 3=GFILE12 # EXECUTION BEGINS 3 TREATMENT UNIT J  3 : SCCIS PINE  E N T E V DURATION COSTS STATE FIRST LAST| MIN MAX |ARJ! VOL |  15 9  6 5 3 2 1 10  I  I 1 1 1 1 | |  41 51 51 53 55 57 61  41 71 71 86 86 86 101  j  j  | | | J \ | |  I I | | | | |  VOLUME CUT |  I I 1 1 1 1 I 1  THE CORRESPONDING NETwOEK HAS  SIM | OUTPUT STATES |  F !  I TI 1 T | T  1 T1 1 T 1 T 1 T  29729 FEASIBLE  9  6 5 10 3 2 1 10 3 2 1 10 1 10 10 10  MANAGEMENT  SEQUENCES  END OF F I L E # EXECUTION TEBMINATED  The  runstream  rates i s l i s t e d a 5 year  to solve  t h e network p r o b l e m f o r 7 d i s c o u n t  below. The p r e s e n t  regeneration  l a g between  $E SOLVE 3=GFILE12 8=-LP 3 1 1 0. 02 5. 3 1 1 0. 03 5. 3 1 1 0.G4 5. 3 1 1 0. 05 5. 3 1 1 0. 06 5. 3 1 1 0. 07 5. 3 1 1 0. 08 5. $END JB *LIST DUGG:DATA.GEN 12 -POL -LP END  S=-POL  n e t worth was c a l c u l a t e d rotations.  PAB=S0L0  with  218  APPENDIX  A Multiple Decision  Thinning  (Goulding's  f i r was  generated  1 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 22  T H E S I S EXAMPLE : THREE 2 1. 1 3 28. 3 30 2. 4 SEND DOUGLAS F I R . S I T E 170 170 200 1.0 1 15 1 2 3 4 5 6 7 8 9 10 SEND  States 20, and  1 - 3  ST0CKI  correspond  P.M.  LEVELS -  OCT.  TwO  -  area,  *  m  61. 6161. 61. 61. 61. 27-  •  • •  -  •  •  •  to i n i t i a l  . .  •m  stand  t h i n n i n g s o f 30% and  and c a n o n l y  be e n t e r e d  a t age 2 0 ) . S i m i l a r l y , basal  area  after  25% and 20%, a f t e r  site  170  05,  ID=PSYU  1976  THINS  m  50. 60. 70.  . . . . .  after  25% state  0 0 0 0 0 0 0 0 0 0 0  3 2 1 0 5 10 4 0 7 10 6 0 9 10 8 0 5 10 1 1 10 1 7 10 1 10 9 10 1 1 10 1  density  o f 600, 900 and 1200 t r e e s p e r a c r e , 5 represent  for  F  1. 2. 2. 2. 42. 52. 42. 52. 42. 5275.  1. 2. 2. 2. 32. 42. 32. 42. 32. 42. 52.  seguences  below:  11:26  OF F I L E  Problem  as a d i r e c t e d graph, f o l l o w i n g the  DATA. GEN 14  ISTING  As A Network  management  management o u t l i n e i n c l u d e d  0.  Problem  Douglas F i r Model).  A set of alternative Douglas  VIII  levels  at  age  r e s p e c t i v e l y . Statas of  the  stand  4  basal  1 (1200 t r e e s p e r a c r e  s t a t e s 6 and 7 a r e t h i n n i n g s  o f 25%  stand  s t a t e 2, and s t a t e s 8 and 9 a r e t h i n n i n g s o f the stand  has been  i n state  1. I n e v e r y  case.  219  the  cut i s distributed  terms  of  diameter  diameter at breast  is  taken  and  20% f r o m Costs  entering  the  defined  o f $50, $60, the  states  f e e 1.50  and of  Thinning  with  is  listed  SHUN SOLVE 3=GFILE14 1 1 1 0.02 1 1 1 0.01 1 1 1 0.06 1 1 1 0.08 $$END  the  $70  600,  30% from  DBH,  In  stand  average  area  removed  |2d1.25, 1.5),  10 i s t h e c l e a r c u t . per 900  acre  and  were  1200  incurred  trees  on  per acre, costs  charged. below present  .08. The y i e l d  optimum management  about  manner.  c o s t s o f $61 p e r c u n i t and c l e a r c u t  the optimal  .02, .01, .06 and  i n t h e same  (DBH) , 50% o f t h e b a s a l  height  $27 p e r c u n i t were a l s o  seguence  the stand  |2d1.0, 1.25). S t a t e  A runstream  of  classes  i n t r e e s o f dbh  respectively. of  across  seguences  that  finds  management  n e t worth, f o r d i s c o u n t t a b l e s and s t a n d  follow:  8=LP 9=FGLICY  the  PAB=SOLO  tables  rates for  MAIN  1  CFCP  SI  *  170  " AGF  STEMS  DISCOUNT  D*H"  PATE  PA  :  VOL  •  *  AFTFP THINNING AV TOTAL  GROSS  * *  *  TOTAL VOL  THINN!NGS  *  P R O D U C T ION  *~'" MAI  *  101. 161.  1601 • 5643.  *  1601.  80  39  3.? 8.A  *  8454.  214  67  224  13.3  241.  11574.  *  14385.  213  01SCOUNT  FATE  16C1. * 4 1 6 2 ^ *  16C1. 6156.  80 195  * *  14847.  227  *  20 31 65  12<" 0 572 1  244  DISCCUNT  :  101. 145.  13.5  27C. :  12853.  *  3.8  101.  1601.  *  1601.  80  *  31  588  6.5  147.  4222.  *  6235.  198  *  15265.  226  *  14.1  DISCCUNT  PATE  20  1200  274. :  T  A  L  -  VOL  96  11.8  75.  2811.  132  9.3  64.  1994.  136  9.2  64.  20C3,  .06  1200 223  0  *  20 67  T  BA  .04  3.8 6.5  P AT F  V.  CfiH  .02  1200 392  20  FI  STEMS  13262.  *  .C8  80'  *  3.6  101.  16C1.  *  1601.  31  5 4 8 "'"6.6  140.  4051 .  *  5952.  189  *"  39  4C8  7.6  147.  5069.  *  8783.  222  *  51  33?  10.1  215.  8981.  *  12696.  247  *  128  S.2  61.  19C1.  80  1C.5  50.  1813.  AGS' T A F I F  CU  VCL  II)  * * STAND  20  22.9  C  C  40 40  36. 1 36. 1  265? 4413  1805 1469  63  45.4  1C928  3926  AG?  TARIF  CU >  20 2 2 . 9  66  VCL IU 2 7. > 11. 0 152 0  182 1 2C76  415 373  44.8 12179  9507  3 2 31.8 32 3 1 . 8  152 4 3 2  0  T.PII:  ril  U D L IU  T_PT. 22.9  CU  20  V O l ^ I L ^ ...... C 0 152  22 32  31.8 21.3  1S24 2163  412 271  68  45.4"12536  10462  A.-C  AGE  TAR I F  20 2 2 . 9  CU >  348  144  92  .32  8  24  56  38  0 64  12  3  e  ** 3 432  4  5 4 348 144  43  0  „ 432  _ 348  8  64  3 432  16  36  714 16<?0 40 3 6 . 1 4 0 3 6 . 1 """3 5 0 2 " 1 4 4 4  0  12  12  40.7  7969  5C58  40 60  4 72  0  8  8  ^  5  144  .  T  „  S  32  0 136 1 56  48 60  0  124  STAND T A B L E 6 7 8 32 92 0 163  44 52  0 68  24 36 """"84 124" "48 4  36  44  68  28 4  8  16  16  4  . S I T E 170 9 10.11  12  13  24  CLASS  C 26  24  20  8  0  12  16 16  CBH C L A S S 14 15 16  17  18  19  20  0  4  16  12  21 22  23  24  25 2 6  27 28  29 3C 3 1  32  33  8 36 4 40 16 44 C 20 40  G  C 8C  ^ DRH  16  24  28 32  A 8L E : S I T E  92  4 ' 5 348 144  0  52  0 172  170 * *  8 20 24 3 28 4 4 28 20  I  20  0  170  Q  24 24  DBH 1  3  ^  1  8  0  ~  8 12  CLASS 5  1  6  ;  ...  7  1  8  1  9  20  21  22  23  24  25  26  •_. 27  23  29  30  31  32  33  40 0 40 8 16 4 8 0 20  1 2 C 2 0  VCL IU 7 . > 11 . 2 C 152 C  1722 2116  136  :SITE  STAND TA SLE 8 7 6 32 92  • * * STAND  346 44C  2 2 21.8 32 31.8  TABLE  24 28  :S ITS 170 9 10 11  32 2C  ** 12  13  8  D EH C L A S S 14 15 16  0 16  20  8  0  18  19  20  21  17  0  4  22 23  16  24 25 2 o  27 23  29 30 3 1  32  33  4 a 40 4C 0 24  4 16 12 28 68  0 20 28 20  20  8 16  8 2C "0 12  0  12  12  16  16  0  0  0 24  '..  0 12  0  0  16  .  .  .  .. ..  3  to • to  222  all  t h e management s e g u e n c e s s e l e c t  per  acre,  area  followed  discount  allowing  rate  by 25% of  the  heavier  for  a  with than  an  average  thinnings  subseguent  .02, a s i n g l e t h i n  Twenty-six c u n i t s of close  utilization  dbh o f 11.2 i n c h e s .  .02, t h e i n i t i a l  inches,  years  c u t ) . At  basal  the  lowest  i s scheduled  a t 39  years.  volume  is  wood  and t h e  when t h e d i s c o u n t  rate reaches  about  cunits  average  dbh o f 10.5 i n c h e s .  the age when t h e s t a n d  of  close  The h i g h  At d i s c o u n t  removed,  t h i n n i n g i s p e r f o r m e d a t 31 y e a r s ,  yields  down  (30% o f s t a n d  greater  r e s p e c t i v e l y . A second  17  o f 1200 stems  rates  volume and a v e r a g e dbh o f t h e y i e l d 9.2  a density  declines to  18  cunits  thinning i s scheduled .08. T h i s  second  utilization discount  i s c l e a r c u t , from  rate  and a t 39  thinning  wood, w i t h also  67 t o 51  an  forces  years.  223  ilPJUJIX  A Wolfe-Dantzig  IX  Decomposition  Program  I n MPSX  PROGRAM('ND') * THIS PECGBAM PEEFORMS A PRIMAL OPTIMIZATION WITH * WOLFE-DANTZIG DECOMPOSITION * * * *  DOUG WILLIAMS APRIL 1974 UBC  * INITIALZ TITLE(•THESIS  DEMONSTRATION')  * * SUBIN READS IN THE PROBLEM NAME AND O E J E C T I V E , AND * SETS UP THE SUBPROBLEM * SUBIN (XPENAME,XCE.3,RATE,ELAG,REPSW,MAXIT,ITER,OBSW ,IOBJ, OPTSW,OBJPSW,RHSPSW,XCHCOL,XCHROW,XPARMAX,XPARDELT) MOVE(XDATA,* TIMERAM') MOVE (XRHS, 'Z') CONVERT PROBLEMS(*PROEFILE•) SETUP ('MAX') MOVE(XDAT2,•TPUNCH') IF (0BSW,0PT1) INSERT OPT1 CRASH PRIMAL NEWZ = XFUNCT XPARAM = 0. MOVE(XOLDNAME,'TIMBRAM2') SAVE (* NAME','SAVEEASF') MOVE(OPZSTART,'CONTINUE*) I F (OBJPSW,PAROEJ) IF(RHSPSW,PARRHS)  *  * START DECOMPOSITION PHASE * DECOM SELECT ('PI',* VOLU1•,V1 'VOLU2' , V2 , 'VOL U3',V3,•V0LU4*,V4, • V C L U 5 ' , V 5 , » V O L U 6 ' , V 6 , • V O L U 7 ' , V 7 , ' V 0 L U 8 ' , V 8 , * V0LU9',V9, 'VOLUA',VA,«PNR1»,D1 'PNR2»,D2,'PNR3',D3,•PNR4•,D4,«PNR5•, #  #  224  D 5 , ' P N R 6 » , D 6 , • P N R 7 * , D 7 , » P N R 8 ' , D 8 , P N R 9 ' , D 9 , « PNRA«,D A , ' •) FREECORE SOBDC (D1,B2,D3,D4,D5,D6,D7,D8,D9,DA,V1,V2,V3,V4,V5,V6, V7,V8,V9,VA,RATE,RIAG,REPS\<J,OPTFLAG, ITER , XDATA, LEI I E 8 LFI1E9,IOEO,XPARAM) IF (OPTFLAG,OOTP0T) CPTFLAG = .TRUE. R E V I S E ( • F I L E » , * REV *,* NOPRINT *) SETUP ('MAX') RESTORE('NAME','SAVEBASE') S E L E C T ( * COL * , ' H C 1 » , H 1 , » H C 2 ' , B 2 , • H C 3 ' , H 3 , » H C 4 « , H 4 , H C 5 » , H 5 , ' HC6•,H6 , ' H C 7 , K 7 , « H C 8 « , H 8 , ' HC9 * , H 9 , ' H C A ' , H A , « H R 1 ' , P 1 , •HR2 ,P2,' HR3 *,P3,* HR4 *,P4,'HE5*,P5,* HR6•,P6,• HR7 *,P7, •HR8 ,P8, HR9 ,P9,*HRA ,PA,' «) XPARAM = OLDP CONTINUE CLDZ = NEWZ OPTIMIZE NESZ = XFUNCT DIFFZ = NEwZ - OLDZ 1 F ( D I F F Z . I E . .001 ,OUTPUT) I F (ITER .EQ. MAXIT ,OUTPUT) OLDP = XPARAM SAVE('NAME*,*SAVEEASE') GOTO (DECCM) 1  f  1  ,  ,  OPT2  SAVEB  *  ,  ,  ,  * OUTPUT SECTION * OUTPUT MOVE(XDATA,'TPUNCH') SELECT{«COL»,•BC1 ,H1,•HC2«,H2,*HC3•,H3,•HC4•,H4,*HC5•,H5, * H C 6 ' , H 6 , » H C 7 » , B 7 , » H C 8 ' , H 8 , « H C 9 ' , H 9 , » HCA« , H A , » H B 1 , P 1 , • H R 2 « , P 2 , « H R 3 « , F 3 , « H R 4 « , P 4 , • H R 5 » , P 5 , • H R 6 • , P 6 , » H B 7 ' ,P7, VHB8« ,P8, 'HR9» ,P9, »HBA» ,PA, » •) PUNCH('VALUE',«LIST«) SOLUTION IF (OPTSW,STOP) IF (ITER . GE. MAXIT,STOP) CONTINUE ,  1  *  * OBJECTIVE PARAMETRICS FOR GOAL PROGRAMMING * PAROBJ MVADR(XDGPRINT,SAVEB) MVADB (XDGNMX,OUTPUT) PABAOBO GOTO (STOP)  *  * RIGHT HAND SIDE PARAMETRICS * PARBHS MVADB (XDGPEINT,STOP) MVADB (XDCPBIM,OPT2) FABARHS GOTO (STOP)  *  * VOLUME DUAL VARIABLES V1 DC{0.) V2 DC(0.)  WEIGHTS  V3  VU  DC (0.) DC (0, ) DC (0.) DC (0.) DC(0.) DC (0. ) DC (0.) DC (0.)  V5 V6 V7 V8 V9 VA * * CAPITAL DUAL VSRIA: DC(0.) D1 DC (0.) D2 DC (0.) D3 D4 DC (0.) DC (0.) D5 D6 DC (0.) D7 DC (0.) DC (0.) D8 D9 DC (0,) DC (0.) DA * DECADE HARVEST DC(0.) H1 DC{0.) H2 DC(0.) H3 EC (0. ) EH DC(0.) H5 DC (0.) H6 H7 DC (0.) DC (0.) H8 DC(0.) H9 DC (0.) HA -it  * DECADE NET REVENUE DC(0.) P1 DC (0.) P2 DC(0.) P3 EC (0. ) P4 P5 EC (0.) DC (0.) P6 P7 DC (0.) P8 EC (0.) DC (0.) P9 EC (0.) PA # * ICC AI STORAGE DC(1000) LFILE8 DC (1000) LFILE9 DC (0. 0) OLDZ DC (0.0) NEHZ EC (0,0) DIFFZ DC (0.0) RATE DC (0.0) R LAG DC (0.0) OL-DP DC (0) IOBJ DC (0) ITER  226  HAXIT OBSW REPSW OPTFLAG OBJPSW fiHSPSW OPTSW  * *  DC (0) DC (.FALSI.) DC (.FALSE.) DC (.FALSE.) DC (.FALSE.) DC (.FALSE.) DC (.FALSE.)  END OF RON  STOP  EXIT FEND  APPENDIX X  Treatment  Onit  Management  Outlines  Demonstration  For Decomposition  Problem  I--_I_E_1____L-__1  E N T R Y  : TO! -  DURATION  OVERM4TURE  COSTS  VOLUME  SIM  S-AIEl£IB_I.L_SIl__L____._lAaE___QL_l--,j: _L_iELAfiL I I I I i I 14 I 1 1 1 F 1 15 | | 2 2 1 -5.0 270.0 F 1 3 9 I 20 I 1 30.0 F 1 8 1 3 20 | 124.0 F 1 3 20 | 7 I 1 10.0 F 1 6 1 40 | 100 I T 1 50 | 4 I 100 I T 1 2 1 40 100 I j T I 60 | 100 | I 1 T i 10 | 80 120 I | T 1 i  THE  CORRESPONDING AND 7 9 5 ARCS  NETWORK HAS 6 2 3 0  I___I_E_I___II____  FEASIBLE  : T 0 2 - OVERMATURE,  QUteUI <I_T_S 15 9 8 7 6 4 10 4 2 10 1 10 4 1 10 1 10 10 10  MANAGEMENT  SEQUENCES  DISTANT  E N T R Y DURATION CO TS VOLUME S I M -E-IEiEI_SI_LASIl__lH__A__l__EA _QL_l__£UT_leLAGL_QULP.UI_SI_rPS 1 I I i 14 1 1 1 1 1 1 F 1 15 | 15 | 2 2 1 5.01 3 1 0 . 0 1 F 9 8 7 9 1 3 30 I 135.0 F 1 6 4 10 I 1 3 I 3 30 I 129.0 F | 1 1 4 2 10 7 I 3 30 I 1 10.0 1 1 F 1 1 10 6 1 40 100 I 3.0| | T I 4 1 10 | 4 I 50 100 | 3.01 1 T 1 1 10 | 2 1 40 100 I 3.01 | T i 10 1 1 60 100 | 3.01 | T I 10 10 I 80 120 I | 2.0| | T I THE  CORRESPONDING AND 1 0 4 0 ARCS  NETWORK  HAS 8 6 9 5  FEASIBLE  MANAGEMENT  SEQUENCES  I_£_I_E_I_U_II_JK_3 : T03 - OVERMATURE, CURRENTLY INACCESSIBLE ENTRY 14 15 9 8 7 6 4 2 1 10  I I 1 1 1 I 1 1 1 1 1  1 1 10 10 15 ' 40 15 40 15 40 40 100 50 100 40 100 60 100 80 120  DURATION  COSTS  30.0 24.0 10.0  -5.0  VOLUME  SIM  I 270.01 I  15 9  8 7 4 10 2 10 1 10 1 10 1 10 10 10 6 4  THE CORRESPONDING NETWORK HAS 4246 FEASIBLE MANAGEMENT SEQUENCES AND 597 ARCS  iaE_I_E_I_lI_T_____  : T04 - AGE 20, HIGH DENSITY  ENTRY DURATION COSTS VOLUME SIM _I_IEI£ia_I_L_ail__I______iAaEA__QL_l__CUI_lELaGl.QULEUI_Sj:_I£S I I 1 15 I 1 I 9 2 9 I 2 I 6 4 2 10 30 6 I 80 \ 4 1 10 40 4 I 80 I 2 10 30 | 2 I 50 10 80 I 1 I 50 10 10 I 60 110 I THE CORRESPONDING NETWORK HAS 316 FEASIBLE MANAGEMENT SEQUENCES AND 151 ARCS  I_EJI_E_I_ii_II____  : T05 - AGE 35, HIGH DENSITY  ENTRY DURATION COSTS VOLUME SIM SlAIElEI_SI_LASIl__X„__A__l__£A„VQL_l__CUI_iELA_L_QUr.EUI_aiAI£S t » I 15 I 1 1 F 1 9 2 91 2 1 F 1 6 4 2 10 6 1 15 55 1 T 1 4 1 10 4 1 20 60 ! T 1 2 10 2 1 30 60 1 T 1 10 60 1 1 20 1 T 1 10 10 I 40 100 1 T 1 THE CORRESPONDING NETWORK HAS 1593 FEASIBLE MANAGEMENT SEQUENCES AND 373 ARCS  I_E_I_E_I___It____ E M  : T 0 6 - AGF. 50, MEDIUM  T R Y  DURATION  COSTS  OEIISITy  VOLUME  SIM  SI_IElEIE._I_L4SIl__i______l_aEA_VaL_l__CUI_lEL_GL_Q_CeUL_S.IAIES 15 8 5 3 1 10 THF  I 1 1 1 1 1 1  1 2 3 4 4 15  I  I 1 1 I I  1 1 2 1 40 1 40 I 40 | 50 I  CORRESPONDING AND 8 0 5 ARCS  |  |  NETWORK  ISEAI_EiJI_!J_II___2 E N T R Y  I 1 1 1 1 1 1  HAS  2331 F E A S I B L E  : T 0 7 - AGE  DURATION  5 0 , MEDIUM  COSTS  SIAT__L£I__T_LASLL__I___A__U^ 15 8 5 3 1 10 THE  I I 1 1 1 1  I  1 2 3 4 4 15  1 2 40 40 40 50  CORRESPONDING AND 8 0 5 ARCS  I  I  1 1 |  NETWORK  Il_AI_E_I_ J_LI___a l  E N T R Y  :  HAS  T08  AGE  DURATION  VOLUME  I  FEASIBLE  COSTS  T T T T  I I I I  3 5 3 10 10  3 10 1 10  SEQUENCES  SIM  1 1 1 1 1 1  6 0 , MEDIUM  I  F | F 1  DENSITY  1 1 2.01 2.01 2.01 2.01 2331  1 1 I I I I  MANAGEMENT  I  1 1 1 1 1 1  I I I  I  F F T T T T  I I 1 1 1 1 1  3 5 3 10 10  MANAGEMENT  3 I  10 10  SEQUENCES  DENSITY  VOLUME  SIM  5IAIElEl--I_LAiIi__l___A__lA_EA__nL_I__CUI_lELAGL _a.t2-I__I4IES ;  I 15 1 8 1 5 1 3 1 1 1 10 I THE  1 2 3  4 4 10  1 2 40 40 40 50  CORRESPONDING AND 8 7 4 ARCS  I 1 1 | 1 1 I  I  NETWORK  I  1 1 1 1 1 1 HAS  I  1 1 2.01 2.01 2.01 2.01 2549  FEASIBLE  1 1 1 1 1 1  F F T T T T  ! I  8  |  5  3 10  1 1 1  3  1 10  i  10 10  MANAGEMENT  SEQUENCES  

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